Before You Start:
(1)-- Basic Idea Behind Thermal Compound:
Thermal compound (paste/grease/hard pads/glue) is used to replace the air that gets trapped between the surfaces of the processor and the heatsink. Air is a very good heat-insulator: even tiny amounts of air trapped in microscopic scratches on those surfaces, can have a big impact on how well the heat is transferred from the processor to the heatsink. The cooler a processor is kept, the better it functions, so it's crucial that the heat (that is produced by the processor) can flow away to the heatsink as fast as possible -- air will stop that flow, so replacing it with thermal paste is needed. But the paste itself isn't a miraclematerial, it will also act as an insulator if you use too much of it.
(2)-- What to look for when shopping for a new cpu-cooler:
First, find out what the exact model of your motherboard(=mobo) is, and then check its manufacturer's site for details: the design of the cpu-cooler must match the design of the processor-socket on the motherboard, so check the sockettype of your mobo to know which cooler to get (for example: socket 754, 939, 940, A, 478, 775, ...). Most cpu-coolers will come with mounting-gear for multiple sockets and they will usually include a tube of thermal paste/grease as well.
Once you've chosen a cooler, you should check whether it will fit on your particular model of mobo. The cooler-manufacturer's website will have a compatibility list for most mobo-models (if not, try Google). There may be components on the mobo that will interfere with the cooler, or the ram memorysticks may block it if they are in slots close to the processor, or the cooler may simply be too big to fit inside your computercase.
(3)-- Tools needed for cleaning & applying new thermal compound:
- Your mobo's manual and other printed information may come in handy.
- Cleaner: best is isopropyl-alcohol (stronger than 70% -- your pharmacist will have it). Also possible: non-concentrated acetone (the stuff you wash your hands with after painting) but make sure it hasn't been polluted and that it certainly isn't concentrated because that would corrode the mobo's printplate. The goal of the cleaner-fluid is to dissolve the old thermal grease -- the old paste will simply disappear after rubbing on it with the cleaner for a while -- and then let it evaporate without leaving traces behind (so use a well-ventilated room).
- Thermal Compound: some companies sell different types of replacement thermal paste/grease, but the difference between them should be minimal. Obviously, if you are going to overclock, then get the high-end stuff. Most coolers will have enough paste included with them (check that though) -- my Zalman cooler came with a little tube that had about 3 portions, and I expect other manufacturers will include a similar amount.
Warning 1: thermal compound stuff can be toxic so avoid all skin/eyecontact!
Warning 2: it usually also has (very low) electrical conductivity, so if you accidentally get some paste onto other mobo-components, then use the cleaner on it, just to be sure that it won't create shortcircuits later on (although it probably won't be a problem). Remember to always give the cleaner-fluid enough time to evaporate completely afterwards. - Applicator:

an ordinary Q-Tip (=for getting the wax out of your ears) is ideal: you can use the cotton-tips to rub the cleaner-fluid onto the old thermal paste, and you can use the plastic bar to spread out the new thermal paste. A little plastic bag may be helpful if you want to use your fingers to spread the paste (remember: avoid skincontact!), and a dust-free cloth may come in handy if you want to wipe some excess paste off.
(1)-- Warning !!:
Static electricity on your body should be avoided as some components can be damaged by that (especially RAM memorysticks/chips), so make sure you don't wear clothing that easily gets charged or could get in the way, and ground yourself every 15 minutes or so by touching a centralheating-radiator or an outer metal part of your computercase.
(2)-- Disconnect All Power:
Have the computer running for a while so the system gets warmed up and current thermal paste gets heated a bit (which will make it more fluid and thus easier to manage later on). Give the computer a few minutes of cooling & de-charge time after powering it down, though, so that the components won't be too hot to handle. Then turn off the PowerSupplyUnit (=PSU) with the switch at the back and disconnect the powercable. All powerconnectors to the motherboard will need to be disconnected as well.
If you're not familiar with the cables and their function, then simply write down what connector they are currently connected to, and how they are connected to it (i.e. which side of the cable plugs into the mobo and the other side into the device). Do this for all connectors+cables if you're not familiar with them -- the mobo-manual will usually have a full description of them all, but it's a good way to familiarize yourself so you can reconnect them again when the job's finished.
Usually, there are two mobo-powercables: one 4or6 pin connector and a big 20or24 pin connector (this may be different), but basically disconnect any connector that is going from the PSU to the mobo.
(3)-- Taking The MoBo Out:
You will need to take the mobo out of the case because most 'aftermarket' heatsinks come with their own backplate (=support-panel at the back of the mobo) and you can only get to that by taking the whole thing out. You'll have to disconnect all the datacables (again: write down which cable goes where first, if you're not familiar with them). Your mobo's manual will likely have a full installation guide, so it's basically that in reverse. You'll also have to take out all the PCI-cards like the graphics card -- note that the graphics card may be held down by a lever so click that open before sliding it out of its PCI-slot (handle with care). The RAM memory sticks can be left in their slots but avoid touching them.
The mobo is attached to the computercase's moboplate with screws so look for them in the corners and in the middle of the mobo, but be careful not to scratch the mobo with the screwdriver and also avoid touching mobo-components with it. You may need to slide the CD/harddisk-drive(s) out of the way a little to take the mobo out -- take care not to bend any jumper-pins on it. Have some clean piece of thick cardboard ready to lay it on so you don't damage the mobo's backside (or your desk). Handle the mobo by its edges and it'll be fine.
You'll notice that the mobo was not directly connected to the moboplate: there are small 'spacers' that sit inbetween. Don't lose them -- just keep them screwed into the moboplate. Their function is to make sure that the mobo and moboplate don't touch eachother via any other part than the spacer to avoid shortcircuits. Don't change their place either -- a mobo has a particular formfactor ("ATX" for example) but the moboplate may have mountingholes for several types of mobo, so a spacer should only be in a mountinghole where a mobo-screw will be.
(4)-- Taking Off The Old Heatsink:
Now that the mobo is out of the case, you should be able to see the processor underneath the big heatsink. Figure out where the heatsink ends and where the processor begins. Also have a good look at how the heatsink is currently holding the processor down so you can try to predict what movements are going to occur as you loosen it.
Okay, this is the risky action in the whole procedure, mainly because most processors still have the [very fragile] pins that connect them to the mobo, attached to them. It's tempting to just pull on the heatsink but 99.99% of the time the processor that's under the big heatsink will be pulled out with it and then you have a 99.99% chance that some of the pins will get bent or damaged in this wrong way of doing it. Note that socket 775 is a LandGridArray (LGA) which has the pins on the socket, rather than the processor, but handle it with just as much care to be sure.
The right way, is to remove the heatsink without moving the processor so that the pins never leave their socket. Most standard cpu-heatsinks have a spring-mechanism that will flip open with some force, so try to control that to make sure that the heatsink doesn't jump up whilst you're opening this (press down on it but not too hard, of course). If the heatsink is attached to the mobo via screws, then the same applies: loosen the heatsinkscrews but try not to move the heatsink itself.
Once it's loose, gently try to wiggle/twist the heatsink around a bit but only a few degrees (like from 12 to 1 and back if compared to a clock-face), certainly not a full circle -- and no vertical movement, just try to slide it across the top of the processor -- you can watch from the side to check whether the heatsink and processor are ready to seperate, or whether the processor might be sucked along if you just pull it up. The amount of wiggling-force shouldn't be greater than what's needed to open the screwtop of a large CocaCola bottle, and I obviously don't mean a screwtop cap that's stuck, but one that just pops open.
If the wiggling works, it will put more and more air inbetween and the heatsink should then seperate nicely after some wiggles without any real vertical force being needed. The processor's socket also has a lockingmechanism so the processor won't really move that easily, but the basic idea is: only minimal vertical force should be used to get the heatsink off the processor -- just keep wiggling until it's ready to seperate, no need to rush it. The fact that the computer was running before you started the procedure, should have heated up the thermal compound, thereby making it more fluid which should also make it easier to move the heatsink around now. If you know it's moving, but can't seem to make it come loose, then try rubbing some cleaner-fluid on the thermal compound that may be showing on the sides and on the gap/borderline between processor and heatsink so it can get sucked inbetween and loosen things up.
If it doesn't move at all, then that's because the thermal paste has hardened a bit too much. It should become more flexible when warm so get a hairdryer and blow warm air onto the top of the heatsink (remember: the heatsink's meant to absorb heat, other components are not -- avoid blowing lots of air onto the memorysticks). It may take a while to heat up the big metal heatsinkblock a bit, though. Keep wiggling it as well -- it's not an exact science, but I think you should be able to tell whether the thing's gonna move or not after 15-20 minutes or so. If it's really not moving, then the heatsink may actually be glued to the cpu, in which case it will probably be impossible to get it off, but this should be very rare (cheap OEM machines may have it).
I've also seen suggestions like these: a very small flathead screwdriver (or thin metalsheet), with the top bent slightly so you can use it as a lever if you insert it in the gap/crackline between cpu and heatsink. It's a possibility, of course, but you may accidentally scratch components that way.
If you happen to pull the cpu out of its socket, then handle it very carefully so that no pins get bent. If some are, then you'll have to carefully bend them back -- which will be a nerve-wrecking experience. Note that each socket has a locking-mechanism, so only place the processor back in its socket when the mechanism is open -- the lever needs to be upright -- check your mobo's manual for this. One of the corners of the processor will be marked with a 'golden arrow' that corresponds to a corner of the socket so that it is placed correctly. The processor will only fit in the correct direction, but make sure you only try that direction, as you may bend pins otherwise.
If you can see that the processor has moved upwards a little but is still in the socket, then just open the socket and the processor should drop back down again by itself ("ZIF / zero insertion force" is needed) -- keep your finger on it while you re-lock the mechanism so that it doesn't jump up again.
(5)-- Cleaning Off The Old Stuff:
The cleaner-fluid will dissolve the old paste, and then evaporate so that no traces of it are left on the processor and heatsink surfaces. Rubbing it in with a Q-Tip may take a while -- like 30 minutes or more, but it's safer than trying to scrape the sticky stuff off, because you risk scratching other components that way. Again, it's a minimal force needed type of approach, so take your time to make sure it's as clean as possible -- avoid touching the cleaned surfaces because the oils from your skin will block the new paste. It may be worth cleaning the old heatsink as well, just in case you ever need it again. When it all looks clean with no sight of the old paste, let the cleaner-fluid evaporate for half an hour or more, before applying new paste.
(6)-- Preparing the MotherBoard:
Most coolers will come with their own gear for attaching the cooler to the mobo, so read the instructions carefully. You may have to replace the "backplate" of the mobo = a small metal or plastic plate at the back that strengthens the mobo around the processor as it needs to support the weight of the cooler (the frontside has some mounting-parts too, that may need to be removed or replaced). You may run into a problem with this, as some of those standard backplates may be glued or fixed to the mobo. The cooler-install-instructions may cover this possibility -- probably advising to keep the standard plate instead, as the cooler will likely have some spring-mechanism to hold it down tightly against the processor anyway. Aftermarket coolers are often heavier than standard coolers and even the recommended weight, but that shouldn't normally be a problem as mobos are quite stiff.
At this point, give it a test-run: put the new cooler on the processor to see how it will fit and if the cooler has a fan, then check in which position it would function best, preferable blowing the heated air out the case as quickly as possible and possibly helping the airflow in the case. Also check how the fan's powercable will be positioned: make sure it won't interfere with the spinning fan once it's turned on.
If you're fitting a heatsink onto a northbridge-mobo-chip, then you may want to check whether the heatsink won't interfere with the graphics card(s), and you may have to bend the heatsink-fins a bit if that will be the case, or even cut a few fins off, but make sure enough remain to take the heat.
(7)-- Applying New Thermal Paste:
Once the cleaner-fluid has evaporated, you can start applying new paste. Use the plastic bar from a Q-Tip to spread out the new paste evenly on the processor's metal heatspreader top-surface. You really don't need much as the paste's only function is to replace the air that would otherwise be trapped in the tiny gap between heatsink and processor. A layer with a maximum thickness of about 0.5 to 0.75 millimetre will certainly be enough, so it's really very thin. Having the paste bulge a little in the center will make sure that all the air is squished outwards when you push the heatsink onto the processor. After setting the heatsink down on the processor, give it a little wiggle to make sure that the paste gets into all the scratches of the contactsurfaces as well. Note that if your cooler has a prefixed thermal pad already, then no extra thermal paste/grease is needed.
Fixing the cooler down may be a rather annoying little task as you will be trying not to move the heatsink around too much whilst trying to screw it down, or locking its springmechanism in place. Each cooler has its own procedure for this, so just hope that the cooler you chose, has an easy way for doing this -- read its instructions carefully.
(8)-- Reinstalling The MoBo:
Your mobo's manual will have a detailed installation guide, and if you wrote down where everything is supposed to go before you started, then it should be a straightforward job. Things to look out for:
- make sure that those spacers inbetween mobo and moboplate, are only in places where you will put a screw to fasten the mobo to the moboplate.
- a good airflow inside your case is just as important as having a good cooler, so make sure that data & powercables are not obstructing that airflow too much. The most common is: cool air in from the bottom-front, and warm air out through the back-top, so make sure enough air can follow that route through the cables (bundle them up a bit if needed).
- don't forget to hook up the new cooler's fan to the right powerconnector and check that no cables will be interfering with that new spinning fan.
- doublecheck everything before restarting the computer, and keep the sidepanel open during that first restart so you can check whether the new cooler's fan is spinning.
----------------------------- CPU ----------------------------------

You can see the fragile pins on the bottom of the processor (left), and on the right, you can see the top heatspreader plate and also notice the 'golden arrow' on the corner.
----------------------------- SOCKET ----------------------------------

On the left you can see that the empty socket is closed because the lever is down. To insert the processor, the lever must be open like in the picture on the right.
----------------------------- THERMAL GREASE ----------------------------------

You can see how thin the layer of thermal paste should be (maximum 0.50 to 0.75 millimetre).
----------------------------- SPACERS ----------------------------------

Notice that the small copper spacers are only in specific mountingholes on the case's mobo-plate, where the mobo will be screwed to the moboplate.
----------------------------- NORTHBRIDGE ----------------------------------


If you're replacing the little cooler on the northbridge chip, then make sure that the graphics card will fit -- you may have to cut a few coolingfins off (or bend them) to achieve that.