Hey. this is nice and easy..
Modem - MODulate/DEModulate - device used on analogue or digital telephony communication lines - i.e. normal phone line or ISDN line
Router - device used on an IP based network to connect different networks together.
Now.. if you are talking about broadband.. a simple USB DSL MODEM will connect you to the BT line and convert the phone signal to a network signal and then the USB connection will connect to the PC (sometimes an ethernet connection is used instead)... to the PC this will look the same as an analog Modem. This is generally used when you have only a single PC to connect to the internet and is a little old now.
If you are in an environment with wifi and more than one PC (xbox, PS3, Wii, laptop, etc, etc) then you will require a braodband router (this will have a DSL modem built in) this will connect to the phone line and then handle all the different communication channels between all the various devices on your network.
Hope that explains a little.. sorry if it is technical but it is tought to break these things down to basic parts without writing a hundred pages...
Let us know if you have a more specific question.. e.g. device to purchase or configuration