How do you know what games you can play online for ps2

As a standard practice game companies include some indication of the game's capabilities on the box. For PS2 titles this indicator is a small box towards the bottom of the back side that says "Online: Broadband Ethernet Adapter Required" or something similar to that. For a general list of online enabled PS2 titles, refer to the following:
http://en.wikipedia.org/wiki/List_of_PlayStation_2_network_games