In the context of Linux, "version" can mean one of two things. It can mean kernel version number and/or issuing entity(e.g. a vanilla sources.org kernel, vs. a redhat kernel or similar), in which case there are dozens. It usually just boils down to 2.4 or 2.6, though, with the former being the older; but highly tested and the latter being newer and more featureful; but subject to greater change. You would probably end up running a 2.6, these days, unless you have a specific reason not to.
The other sense of "version" is "distribution". There are hundreds of distros; but a relatively small number of them matter much. One usually speaks about them according to package manager, as that ends up defining the experience. There are the Apt based distros, my personal favorite of the moment. Debian is the big name there, as they brought Apt to the scene, and have a huge set of packages, and lots of supported architectures. Their quality and stability are also well regarded. However, they aren't known for a snappy release schedule, and they are, therefore, more popular with server, embedded, and slightly old-school workstation users, than with hip-'n-flashy desktop users. Ubuntu, based on Debian and Apt; but released more aggressively and with the latest features, is the new new thing among desktop users. Both of these distros are quite good, which one you want will depend on what you wish to do.
The RPM based distros, of which RedHat is the iconic example, are the other main option. People often don't like RPM as much as Apt, though it has gotten better in recent years; but these have the advantage of being more "corporate" which means a certain degree of polish, standardization, and support that the indie projects don't always have(though the indies have their own advantages, and I happen to like them better). SUSE is the other big RPM based corporate player, and is generally well regarded. Fedora is RedHat's community beta, more or less, and so is the free version that many RPM users run. OpenSUSE is the SUSE analog to Fedora. There are also, because of the GPL, distros like Whitebox, which are recompiled versions of the commercial releases, with all proprietary content stripped out; but with pretty much full compatibility.
The Portage based distros are relatively scarce. Gentoo is the big player here, which is appropriate as they invented the system(well, some might say adapted the idea from the *BSDs; but that's another issue). Gentoo emphasizes customization and bleeding edge features through custom compilation. There are a few distros based on Gentoo, mostly designed to make it easier to install.
Then there is the "other" category. Distros like Slackware and Linux from Scratch, among others, that aren't defined by one of the big packaging systems. Often very popular with a certain userbase; but not as common as the big two Apt and RPM.
Take a look at Distrowatch.org for more concrete details.
For your purposes, I'm not quite sure what to recommend. You'll probably want a fairly modern, not too difficult desktop oriented system. Ubuntu would be my off the cuff recommendation. I would warn you, though, that while desktop Linux is really quite good these days(I'm on my second year of using that alone, and on a laptop, and maybe 5th year of making decently heavy use of it), it isn't the gamer's platform of choice. Some games, mostly those based on Id engines, or the Unreal engine, have native binaries. These aren't so hard to get working, assuming your graphics card works(which isn't hard if you have an NVIDIA, can be a dodgey business with an ATI, though they do suck less than they used to). Other games that only come in Windows versions can be run with WINE, a package that you install on top of Linux, or Cedega, the commercial version of WINE, designed for game support. Sometimes this works just fine, other times it doesn't work at all. Sometimes it is just in between. It isn't hopeless by any means, and all the trends are in the right direction; but gamers will probably be the last people to be wholly satisfied with Linux.
I recommend: Start by downloading and burning some LiveCDs, from various distros. This'll cost you nothing but bandwidth and blanks, and give you an idea of how Linux feels, and how well it plays with your hardware(probably pretty well; but it is better to have a nasty surprise with a LiveCD, than halfway through an install). Next, either set up a dual boot or secondary HDD on your main box for Linux, or load it on an older secondary box. Play with it, as much as you can, get to know the software. This will allow you to move ahead at your own pace, with no nasty surprises. If you get to the point where you feel comfortable going all the way, I heartily encourage you to do so. If you continue to need the Windows partion for a few programs and games, no big deal. Have fun with it.
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