I'm not going to go into great depth here about the technical aspects of each distro, but rather a general overview. There will be links to each distro embedded in the article. Based on my experiences, I will also list recommended memory. Here are some of my favorite lightweight distros:
Puppy Linux is one of my all time favorites. The Puppy linux community is very active and vibrant. Many members have spun their own versions called Puplets. Puppy is very easy to remaster and create your own variant. Some people have taken basic Puppy and stripped it down to the bare essentials, and others have added apps and features to make it very robust and complete. Recommended memory- 96mb+ Screenshots
A great straight forward approach to lightweight linux. Does not come with many apps preinstalled, but that is part of the beauty of it. Recommended memory- 64mb+ Screenshots
The legendary distro known as Simply Mepis, decided to throw it's hat into the lightweight distro arena. And what a great job they did. Using the Fluxbox desktop environment, it is a bit heavier than either Puppy or slitaz, but still light enough to be included here. Being based on the venerable Debian, you are able to download a vast array of apps and tools from Debian's repositories. Recommended memory- 192mb+ Screenshots
Tiny Core Linux is an absolutely amazing achievement when you consider the base .ISO file is only 10mb. That's not a typo, you get an operating system with GUI in only 10mb. It is exremely minimal as you might have guessed, but it's easy to install apps through it's tiny package manager. Recommended Memory- 32mb+ Screenshots
There are other lightweight distros out there, but these 4 represent what I think are the heavy hitters of the lightweight distro world, and will extend the life of that old computer gathering dust. Check out the screenshots and download away.