Playing against the computer vs. playing interactively against other people
The former is faster, more under your control, but not very social
The latter can be very slow (for both technical and human reasons), but it is much more of a community activity
Playing card games or board games vs. arcade games vs. adventure games
Card and board games can be played either against the computer or interactively. They typically mimic the "real" versions and include everything from Go Fish through Bridge and from Checkers to Backgammon
Arcade games are usually played against the computer and often require significant physical skills. Arcade Games Online provides access to dozens of them
Adventure games are almost always played interactively (that is, against human players who may be anywhere in the world). They are often quite involved and may require many weeks of play to complete a game. A good source of information and links is The Adventure Games Coalition
If you don't know the rules, they are usually available in help files on the gaming site. Or you can use a specialized search engine called EveryRule.com to access the information you need
WARNING: Playing games on the web can be highly addictive, but, if you're still interested, browse to Game Links