An ADSL modem, is a device that communicates via an ADSL enabled telephone line with your Internet Service Provider and transmits and receives data to your computer.
An ADSL Router is actually a combination of ADSL modem and Router. The modem part works in the same manner as above, but the router is capable of routing the data to different computers. So you could have two or more computers connected to the router all capable of internet access, and the router sends each "packet" of data to the correct computer. In other words, you can have one internet access point, but many machines connected to it (via the router).