How do I find slimes in Minecraft?

Edit: As of Minecraft 1.5 or so, Slime spawning has changed yet again. Right now, Slimes will only spawn in certain chunks of your map, depending on the map's seed. There are tools online such as this one that let you input your seed, and it will output the chunks of your map that will spawn Slimes.

Slimes spawn within 16 blocks of bedrock, in any light conditions. I think they were removed in the 1.2 Beta, but it is confirmed that they have returned in 1.2_01.

Slimes spawned fairly frequently back in Alpha, but I believe Notch reduced the spawn rates by quite a bit since then. The 1.3 update increased the slime's spawn rate yet again.

The fact is that slimes now spawns naturally at night (technically light level < 8) in swamp biomes ("Swampland" in debug screen) with y-coordinates ranging from 51 to 69. This feature is added in 12w40a development snapshot or 1.4.2 (Pretty Scary Update) release, making them much easier to be found.

Slimes also spawn below y-coordinates of 40 in specific chunks. Minecraft Wiki has a formula on the slime spawning mechanism and there are numerous tools to find slime-spawning chunks, however since it is subjected to change, it may be far easier and better to find slimes by simply look for swamp biomes on the ground.