You can access lots of free games on the internet, including games you play on-line and games you can download to your computer.
On-Line Play
There are lots of sites where you can play on-line games. However, on-line games aren't as elaborate as games you download or buy and they can be slow if you're using a dial-up connection.

Some of my favorite on-line games sites are:

Alfy
Cartoon Network
Disney On-Line Games
Free Arcade
Fun Brain
Fun School
Kiddonet
Lego Games
Neopets
Nick Jr.
Pogo
Prongo
Speedway Showdown
Stagecast
Yahooligan's
Zeek Games

Downloads
You can download games off the internet and run them on your computer.

Download.com has lots of free downloads.

Trial Versions
Trial versions are games you can download and play on your computer, but they usually expire after a specified time or they are only a sample of the real game. For example, the demo version of Zoo Tycoon only has 4 animals compared to dozens in the full version.

Trial versions can sometimes be found on the software manufacturer's web site.  If you want to try a game, find out who makes it and then check their web site to see if they have a demo version.

Trygames.com has hundreds of games you can download for free and play for 20-60 minutes. Then if you like the game, you can buy it without having to download again.

The following sites also have free demos and trial versions:

Apple Game Demos
Electronic Arts Demos
Microsoft Game Trials
PC Game Demos
PC Gameworld Demos