Look on the back of your games. Those with the blue "Nintendo WFC Connection" can be played online. Any games you have with online components should be explained pretty thoroughly in the manuals. Not all games can go online. Some quick google searches on top of the Wikipedia links provided could help you find online Wii games. I always recommend reading reviews from a reputable site like gamespot.com to figure out if you are thinking of buying a good game or a dud. Most good game review sites will include reviews of how good the online features are in each game.
As PP mentioned you can't play the same game with others on different gaming systems (Xbox, etc.)
Most games for Wii do not have voice chat, b/c Nintendo views themselves as very family friendly and don't like to facilitate environments where kids could be cussed out and in general verbally abused by complete strangers, which can happen on a lot of online games. Usually players have to use pre-chosen phrases or type. Often to play online people will have to exchange "Friend Codes", another method used by some games to prevent strangers from making contact.