I've been invoked, and nobody told me!
TR user jmgarvin said I like Debian, and he'd leave the description of it to me. I'll start with that, then, though I'll make comments about others as well.
[b]Debian[/b]: Debian is an expert system administrator's wet dream. It's LSB- and LFS-compliant, stable as can be (even in the up to date Testing and Unstable versions), has the largest core software package repositories of any distributions out there (at close to 20,000 packages), and has the best package management system available (try convincing me otherwise) called APT, so good in part because it's so scriptable. It is not the easiest distro in the world for a rank Linux newbie to grasp, however, and if you want to try this distro first I recommend getting help from someone familiar with Debian. This is the second oldest Linux distribution still in existence, so it has had a very long time to mature, is committed to "free" (as in speech) software, and has the largest community support base of any distribution.
[b]Fedora Core[/b]: This is the free/community version of the Red Hat distribution. It is intended to be somewhat cutting-edge, so you might run into some occasional package stability/incompatibility flake-outs from time to time. It is afflicted by the Red Hat way of doing things, which means it has a slightly nonstandard kernel, and is not exactly LSB- and FHS-compliant. It is geared toward use of GUI tools for administration in such a way that command line stuff suffers sometimes (for instance, have fun futzing with a bunch of separate Ethernet interface config files buried five levels deep in the filesystem within a scripts directory, for some bizarre reason, if you want to fine-tune network configuration by hand). Its primary package management system is YUM (adopted form Yellow Dog Linux), which is very capable but is also slow, and tries to do too much for the user at one time. There's an RPM version of Debian's APT available, as well, though its use is deprecated in favor of YUM.
[b]Gentoo[/b]: Like Debian, this is a purely community-based distribution. It is designed to be managed completely from source code, using some automated compile-and-install tools to make the job manageable. These tools are extremely slick, and there's something to be said for the fine control you get over system configuration in this manner, but ultimately it's not all that much more fine-tune-able than Debian and Slackware and it's difficult to justify spending three days compiling and installing your OS and its operating environment software. At least, I think it's difficult to justify. It's certainly not something you'd want to use for the standard OS of an Enterprise network: the administrative overhead would bring your organization to a grinding halt. For a single hobby machine, however, many people swear by it. Oddly enough, the original creator of the Gentoo distribution is now an employee of Microsoft. I don't know what that means for Gentoo.
[b]Knoppix[/b]: This is the original, and canonical, LiveCD distribution. That means you boot from a CD and run the OS entirely in RAM, then take the CD out when you're done. Nothing has been installed on the hard drive. There's about five hundred different variations on Knoppix for different purposes out there. Knoppix is very good at being a LiveCD distro (it detects and handles different hardware types quite well), but is clunky to install and eventually ends up having software issues if you try upgrading software on it much after installing it to the hard drive. Use some Knoppix version as a portable LiveCD OS if you like. I don't recommend it for hardware installs.
[b]Mandriva[/b]: This is what happened when Connectiva and Mandrake combined, with Mandrake being the most recognizable influence. I hadn't really used Mandrake much personally until I got my current job, but before that I did have a vague sense from my occasional dealings with it that it wasn't what I wanted. It has a reputation for being even more flaky than Fedora, and without the excuse of really being cutting edge, for instance. It's also very GUI-centric, to the extent that it is sometimes quite bad at CLI-based things that every other Linux distro (with rare exceptions) is good at. It also inherits some of Red Hat's screwed up system configuration decisions, having evolved from sort of a Red Hat knock-off. You'll almost certainly see a great deal of similarity in default system configurations under the hood between all RPM-based distributions, since the RPM package system was invented by Red Hat in the first place, and Mandrake is definitely an RPM-based distro. Now that I've started having to deal with Mandrake far more often (more than half the Linux systems in the company are running Mandrake 10.1), I not only have a vague sense of dislike: I actively loathe it. The urpmi package manager is even worse than YUM for trying to do too much for you, and it even fails to do some things entirely. It's also damn near impossible to find good documentation for a lot of stuff on Mandrake systems, including how to use urpmi and related package tools. It really drives me up the wall. I got totally spoiled by the wealth of manpages in Debian, for instance, and Mandrake is at the other extreme. In fact, one of the Mandrake users at the company recently started asking me if there was some way to get more and better manpages on his computer: I'm not the only one that notices that Mandrake has manpages for, comparatively, almost nothing. Yeah, it's easy to install. The default desktop is very clicky and colorful. Beyond that, my honest opinion is that it's among the worst of the major Linux distributions.
[b]OpenSUSE[/b]: This is the community version of Novell's SuSE Linux. I don't have any experience with OpenSUSE, but I rather expect it will be a lot like SuSE Linux Enterprise Server and the like, just without the proprietary add-ons. My experience with SuSE is mostly with pre-Novell versions. It's an easy distro to install, as long as you don't try to customize anything too much: just go with the options it presents most clearly, and you'll be fine. It's very much a kitchen sink distro, in that it'll include the kitchen sink in your install. The GUI management tools are very slick, but the CLI admin tools rather leave something to be desired. Novell is like an eager, naive kid in the Linux world, wanting to get involved with everything from development to marketing for Linux in general, and even seems to be willing to lose the occasional customer if such is necessary to get that customer using Linux in general rather than going to Microsoft Windows or Sun Solaris (for instance). It's all very endearing, and I rather hope that Novell SuSE continues to become the powerhouse enterprise Linux distribution it seems intent on becoming.
[b]Slackware[/b]: This distribution is the oldest still living, even a couple months older than venerable Debian. It's maintained by one man, and is very bare-bones. You can install anything you want on it, of course, as long as you're willing to download, uncompress, compile, and install everything yourself, starting with tarballs offered by the myriad of separate open source software project maintainers in the world. Well, that's not entirely fair: Slackware does have a package manager of sorts (don't bother using it), and it does come with a big ol' pile of tarballs if you download the complete ISO set to burn to CD for installation. This is very much an old-school bare-metal hacker's distribution, though. When someone says Linux is a "hacker's OS", they mean Slackware.
[b]Ubuntu[/b]: This is a very "user friendly" Debian-based distribution. It doesn't act much like Debian generally does, though. For one thing, Ubuntu doesn't even have a root account in a standard install. It drives me nuts. It's probably wonderful for people coming from Windows who just want a working computer and don't give a rat's ass [i]how[/i] it works.
[b]Yellow Dog[/b]: I mention it only because it was referenced in the description of Fedora. It's Linux designed specifically for installation on the Mac architecture. You're probably not in need of this distro, especially since there's a Mac version of Debian available, too. Heh.
There's more, but that should tide you over for a while.
Oh, yeah, and here's where you can download Debian install ISOs:
http://www.debian.org/CD/netinst/
(the network installer CD of the Stable release, for use with a broadband connection)
http://www.debian.org/devel/debian-installer/
(the network installer CD and the full CD set of the Testing release, for use with a broadband connection)
http://www.debian.org/CD/
(a whole slew of options for acquiring Debian installer CDs)
In general, you'll probably want the "i386" architecture version, since that's for the "PC compatible" type computers (aka "IBM compatible", aka x86, and so on).