General

All but the slowest PCs can now run original arcade games. Download an emulator, a ROM (the actual game code from the machines) or two and you can play the games as much as you want without requiring a pocket full of change.

Some emulators emulate just one game, most emulate many as the hardware for different games was often similar. Here are some of the best sites to check out.

Retrogames and Vintage Gaming Network - Probably good sites for the latest arcade emulation news. Updated regularly.
The Vault - Gaming the the way you remembered it.
Emulation Unlimited - Large site which hosts lots of cool emulation related sites too.

MAME

The Multiple Arcade Machine Emulator. Emulates around 4500 working games! Versions available for just about every platform imaginable including DOS, Win32, UNIX and the Amiga! Emulates all the classics from Space Invaders though to the likes of Mortal Combat. Also emulates the SNK NeoGeo console: Metal Slug and Metal Slug 2 are amazing! The best place to get the games are newsgroups: alt.binaries.emulators.mame

Official MAME - Updates on the current status of the project, and news on upcoming supported games.
Mameworld - One of the best MAME resources.
Mame32qa - Download the Windows version of MAME.
ClrMAME Pro - Manage those thousands of ROMS with this ROM mamger.
Redump.de - Optimised MAME builds to download.
Romshare - Loads of ROMs to download.

Other

Raine Homepage - The Raine Emulator now emulates well over 100 games including a few crucial ones MAME does not. Specializes in Taito and Jaleco Games.

There are many other good emulators. Developement on many has now stopped ue to the MAME onslaught. Worth a mention are:
Callus - This emulates mostly Capcom games like Street Fight 2 and Strider.
System16 - System16 emulates including games like Outrun, Shinobi, Super Hang-On and Golden Axe.
Impact - Newer 3d Arcade Games. Findout about them from the links above.