I’ll try to keep this simple:
A "modem" is what connects your computer to your internet supplier and basically converts the signals on the DSL to a computer network connection.
A "router" allows the network connection coming out of the modem to be shared by several computers.
You can connect your computer to the above devices either by using a network cable ("wired" – fastest data transfer rates, most secure) or, if your computer has a wireless network card, wirelessly (less secure, but convenient to set up).
What they’re calling a "wireless modem" is basically the DSL modem + a wireless router built together.
If the modem your internet supplier gave you doesn’t have an antenna in the back, and only has one network connector in the back, then if you want to connect more than one computer to the internet at a time, then you’ll need to buy a router. You can buy a router almost anywhere now-a-days (here in the US they sell them at Walmart and Radio Shack) and can be had for less than $100USD. Just about all the cheaper ones offer both wired (Cat5e connectors) and wireless (2.4 GHz) connections. If at all possible, I recommend you use a wired connection to your router. It will offer the greatest level of security and performance. If you must use a wireless connection to your router, then I highly recommend you google for information on how to set up the security settings on your computer and router.
Hope this helps…