The functions of a router and a hub are quite different from one another, even if at times they are integrated into a single device. The hub serves as a central connection for all of your network equipment and handles a data type known as frames. Frames carry your data. When a frame is received, it is amplified and then transmitted on to the port of the destination PC.
In a hub, a frame is passed along or "broadcast" to every one of its ports. It doesn't matter that the frame is only destined for one port. The hub has no way of distinguishing which port a frame should be sent to. Passing it along to every port ensures that it will reach its intended destination. This places a lot of traffic on the network and can lead to poor network response times.
Additionally, a 10/100Mbps hub must share its bandwidth with each and every one of its ports. So when only one PC is broadcasting, it will have access to the maximum available bandwidth. If, however, multiple PCs are broadcasting, then that bandwidth will need to be divided between all of those systems, which will degrade performance.
Routers are completely different creatures. Where a hub is concerned with transmitting frames, a router's job, as its name implies, is to route packets to other networks until that packet ultimately reaches its destination. One of the key features of a packet is that it not only contains data, but the destination address of where it's going.
A router is typically connected to at least two networks, commonly two LANs or WANs or a LAN and its ISP's network (ex. your pc or workgroup and EarthLink). Routers are located at gateways, the places where two or more networks connect. Using headers and forwarding tables, routers determine the best path for forwarding the packets. Routers use protocols such as ICMP to communicate with each other and configure the best route between any two hosts.
Today, a wide variety of services are integrated into most broadband routers. A router will typically include a 4-to-8 port Ethernet switch (or hub) and a Network Address Translator (NAT). In addition, they usually include a Dynamic Host Configuration Protocol (DHCP) server, Domain Name Service (DNS) proxy server and a hardware firewall to protect the Local Area Network (LAN) from malicious intrusion from the Internet.
All routers have a Wide Area Network (WAN) Port that connects to a DSL or cable modem for broadband Internet service and the integrated switch allows users to easily create a LAN. This allows all the PCs on the LAN to have access to the Internet and Windows file and printer sharing services.
Some routers have a single WAN port and a single LAN port and are designed to connect an existing LAN hub or switch to a WAN. Ethernet switches and hubs can be connected to a router with multiple PC ports to expand a LAN. Depending on the capabilities (kinds of available ports) of the router or hubs, the connection between the router and hubs may require either straight-through or crossover (null-modem) cables. Some routers even have USB ports, and more commonly, wireless access points built into them.
Some of the more high-end or business class routers will also incorporate a serial port that can be connected to an external dial-up modem (useful as a backup in the event that the primary broadband connection goes down) and a built in LAN printer server and printer port.
Besides the inherent protection features provided by the NAT, many routers will also have a built-in, configurable, hardware-based firewall. Firewall capabilities can range from the very basic to quite sophisticated. Among the capabilities found on leading routers are those that permit configuring TCP/UDP ports for games, chat services and the like, on the LAN behind the firewall.
So, in short, a hub glues together an Ethernet network segments, a router can do those functions plus route TCP/IP packets between multiple LANs and/or WANs; and much more.
I hope that's made things clearer for you