Ah, the question asked by scores of people everywhere. Just getting Linux in the first place can be a huge hurdle when you're not even sure which variant to get.
Let it be understood that for all the fuss and debate over them, the different distros of Linux are really not that different from each other; In the end, they really are just somewhat different packagings of the same underlying operating system. Certainly, one may have a couple of added features which the other does not have, and others may have a prettier GUI or handier set of utilities, but Linux is still Linux.
That said, the decision will be easier if it is made multiple-choice. To this end, I will simply list the most important distros of Linux, and leave it up to the end user to make his or her own decision about which to choose. Note that these are only the most major distros, and there are dozens (if not hundreds) of minor ones which could not all be covered in a single webpage. (Well, in theory I guess they could be, but they won't be here.) I have added what comments I am able to provide, but I really haven't used a lot of different distros, nor have I analyzed them to the point where I could provide much useful advice on which one to pick. So basically, just flip a coin or something, I dunno.
The original Linux distro, Slackware has a rich and storied history. (Inasmuch as a version of an operating system that is only about 10 years old can have a rich and storied history.) It is my opinion that Slackware is unfairly beat up on; People call it unreasonably difficult to use, because it suffers from lack of beautification techniques to make things easier on the user. Slackware is the distro which I personally use, and I have found it quite user-friendly in the several years which I have used it. I like its system of package management; I find Slackware's pkgtool to be much easier to use than, say, Debian's dpkg. Slack may not have the hand-holding tactics employed by The Great Red Hate (what I use to refer to Red Hat), but that's a good thing.
The Microsoft Windows of the Linux world, Red Hat became enormously popular for the same reason Microsoft Windows did: People like things which are easy-to-use. Red Hat covers up the operating system for the user, hiding it behind a system of menus and mouse-operated environments. And while that sort of thing may make it easier to use, doesn't that really defeat the purpose of installing Linux in the first place? People install Linux for the direct control and power it gives them. I liken Red Hat to buying a Corvette and then installing a tiny, underpowered engine inside it for fuel efficiency. The combination just doesn't fit. If you are truly tempted to get Red Hat because ease-of-use is a serious concern for you, I would definitely recommend you get Mandrake Linux, a sub-distro of Red Hat, as Mandrake incorporates even more things which make Linux easier to use.
I installed and used Debian at one time. To be honest, I wasn't very impressed. It's a capable system, but it's bogged down by an amazingly cumbersome package-selection-and-installation utility, dpkg. (There's also something called dselect, which is supposed to be easier to use but which I never attempted.) I didn't use Debian much so I can't comment beyond that.
I have never used SuSE, but I've heard good things about it. This is the German Linux, and the most popular Linux distro in Europe. Apparently it comes with a lot of extra tools to make it more powerful. This is a good thing, because Unix/Linux as a whole is powered by the commands and programs it contains. The more proggies you have, the more powerful your OS. If you want to step off the beaten path and use a semi-alternative distro, SuSE is worth a look.
Aimed at the server market, TurboLinux is, as the name suggests, the "power" version of Linux, for people who want to put their computers to business or industrial use. If this is what you want to do with your computer, by all means, get Turbo; It is a solid system.
What ever happened to this distro? At one time, it was fairly significant. Now, I never hear anybody talk about it. Did Caldera go out of business or something?
Uh... No comment.