If you want to use wireless (which can be very tricky) you can get a wireless game adaptor and connect it to the Ethernet port on the back of the PS2. They run about $80-90. A USB adapter will not work on a PS2.
An easier way would be to use two HomePlug powerline to Ethernet adapters. You can get these for $30 each at many Internet reatailers, for example: http://www.newegg.com/Product/Product.asp?Item=N82E16833173007
You simply connect one to a LAN port on your router and to an electrical outlet nearby. Then connect the other to the Ethernet port on the PS2 and to an electrical outlet nearby. The PS2 is now connected to your Internet connection.
i already have a good 802.11g wireless router, but what do I need to take a SlimLine PS2(with network adapter) online - a standard USB plug-in thingy, or some sophisticated thing that costs a lot of money? i need help! mind, a 13-year-old's allowance only goes so far, are there any ways sub-$50 to take it online? Battlefeild 2: Modern Combat is begging to be played in its prime!!!

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic