Tiny Core Linux. The name immediately hints at the creator's intentions, but only once I had booted up the LiveCD did I really know how "Tiny" it was. With the .iso file (available here) being about 10 megabytes, one could easily download Tiny Core 3.5 over dialup!
Once you start up Tiny Core, the first thing you will notice is the incredible speed. This does "cost" the user a web browser, GCC, games, and Office Software, but preinstalled software really isn't the focus of this distribution. All you get for now is a Macintosh-like dock with some basic system utilities. Tiny Core really does take minimalism to heart, but not to the point where it's impossible to learn how to use it.
(What you see when you first boot into the LiveCD)
Installing TinyCore to the hard drive is no easy task. To start up the installation, click on the "Panel" icon on the dock, then click on the button that says "HD/USB Install". One has to know what they are doing here, because the installation involves partitioning. If you don't, I'm sure you can check the FAQ or the forums for help. If you do know what you're doing, you can go ahead with the installation, but be warned that at times it seems to refer to flash drive (which it assumes you are installing to) when you should enter the information for the hard drive instead. After that, it should install fairly painlessly, at which point you can boot into Tiny Core from your hard drive with ease.
Once you have it installed, there is not a lot to do. To install an application, simply click on the dock item called "Apps", find the software you want to install, and click the "Go" button near the bottom of the apps window! From here on you can do whatever you want with it. You can make it into a server, a fully functional desktop OS, or just a web browsing platform. With thousands of applications available, Tiny Core can be made into just about whatever you want it to be.
Because of the minimalist nature of Tiny Core, it will only be whatever you make it, and nothing more. This is the true elegance of Tiny Core. It's a truly blank slate, leaving the user with complete control of the destiny of their machine, instead of relying on somebody else to tell the user what they need when they boot up.
There's not much more to be written about this distribution, mostly because there's not much to it beyond what you, the user, decide to do. It is about as close to building your own distribution of Linux as one can get without actually doing it, and it certainly isn't for absolute beginners to Linux. But if you're a seasoned user of Linux (or even just Unix in general) and you're looking for a distribution that you can customize to a specific purpose, or you just don't want to sacrifice speed for "user friendliness", you should certainly look at Tiny Core.