The cable or DSL modem's job is to convert the incoming media of a coaxial cable or telephone line into a standard 802.11 Ethernet port. (That's also one of the purposes of the Optical Network Terminal [ONT] in Verizon's FiOS offering.) Typically, that one port is also assigned the single (static or dynamic) IP address that your ISP allots to your home or small office. The router's job is to share that one connection among potentially many devices on your LAN. It determines what network traffic should be routed from a device on your internal LAN to a device on the Internet and vice versa. I've often heard the anology that the routing function is like a traffic cop directing traffic. Really more analogous to a security guard at a private residence. If the router receives a network packet from one of the devices on your LAN that has a destination IP address that's outside, it will pass that packet through to the WAN or Internet. On the WAN side, the router will examine the packets it sees and if the packet matches the WAN IP address it is set to, it will pass the packet to the LAN (with some "adjustments" we will get into below) if there is a device on the LAN expecting it. The primary task of a router is to act as a gatekeeper or gateway.
Generally, residential (home) customers of an ISP are allotted only a single public IP address. That IP address may be issued dynamically using a DHCP server that the ISP maintains or it may be assigned statically. That single address only supports one networked device. The question then is how do you share this single IP address with more than one networked device from your LAN? You need some way of making all the computers and networked devices in your home appear to the outside world as a single device. Another task of the router is to do just that. (At least, that's the method used by most home and small office networks.)
I've seen several forum posts asking some variation of the question, "I only have a single computer to connect to the Internet. Is there any benefit to having a router?" In my opinion, the short answer is "Yes." While there is no technical reason why a single PC can't be hooked directly to the Ethernet (or USB) port of a cable or DSL modem, generally not a good idea. If you purchase, install, lock down, and monitor your PC using a good software firewall, connecting a PC directly to the modem may be fine. The problem is many people don't understand what it means to "lock down" their PC and leave too many ports open. Even fewer monitor their firewalls as they should. Microsoft's own Internet Connection Firewall has improved in the since the Windows XP Service Pack 2 release. Still, it often doesn't have the configurability and capabilities of a hardware router firewall. Commercial firewall software packages are better, but their cost is comparable to buying a hardware firewall router.
The table below lists some typical wired Ethernet routers. The Wireless Networking Equipment section already listed some typical wireless Ethernet routers. The routers listed below differ in only that they do not have built-in WAP capability. These are becoming rare nowadays as the prices of the wireless-capable routers have dropped to match those without and the popularity of wireless networking has increased. Mostly, I see these used in small businesses that don't want the risk of having an active wireless network. (It's also possible to buy a wireless router and just shut off the wireless capability.)
The Linksys BEFSR41 shown here is a cable/DSL router/firewall with a built-in 4-port 10/100 switch in the back for the LAN side. Linksys also makes a couple models with built-in wireless access points (WRT54GS), a single LAN port, and an 8-port model. The router is really only needed if you are planning to share an Internet connection.
The Netgear DG834N Rangemax is a router like the BEFSR41 above, but also includes a built-in DSL modem and an 802.11n wireless access point. Devices such as these take the place of a separate cable or DSL modem (like those shown in the previous section) and router.