is a device that MODulates & DEModulates data - it takes digital information and converts it into analogue form in one direction and does exactly the opposite in the other direction. It allows your computer - which is a digital machine - to communicate via a telephone line, which handles analogue information.
A router, on the other hand, does a different job. It acts as a sophisticated switch that knows what is connected to it at any given moment. It takes data packets from one network - say the Internet - and decides on the best ROUTE for them, based on the state of the network with which it's dealing at the time. A router maintains a complex map (for want of a better word) of the state of a network, and sends the data to a given point in the most efficient way. Routers are used out on the Internet to decide how best to route Internet traffic from point to point.
In your home, a router makes an ideal interface between your home network and the Internet. Many ADSL routers (but not all of them) have ADSL modems built in. Those that don't have their own modem will have a port into which you can plug an ADSL USB modem. Routers will have RJ45 Ethernet ports for CAT5 network cables, and if you want to have a wireless network you'll need to think about a wireless router/modem combination, or a wireless access point - both will require all computers to have wireless network adapters - either internal PCI cards or external USB adapters. Laptops can use PCMCIA card wireless adapters.