Fedora is a Linux distribution from Red Hat that doesn't cost any money, includes only open-source software, and always includes the latest versions of software.
Fedora is a community-supported project. That means that you can be involved in creating Fedora, if you want.
A new version of Fedora comes out every six months or so. When a new version comes out, the previous version becomes unsupported about a year later. This means re-installing your OS every 6 - 8 months (currently upgrades between Fedora versions don't go so well).
The advantage of using Fedora is that you're always on the cutting edge of Linux development, more so than any other major distro, while still having a stable operating system.
Red Hat Enterprise Linux is based on Fedora, so if you want to learn a little about RHEL for free, use Fedora. RHEL6 was based on Fedora 13.
What is the difference between Fedora and Red Hat Enterprise Linux? Well Red Hat Enterprise Linux is officially supported by Red Hat for 7 years. It's a better choice if you're running important machines in production, and you don't want to upgrade your OS all the time. It also has many nice features that are useful if you are running more than a few servers.
Also, you can (sometimes) actually call Red Hat and get support for RHEL, which you will never be able to do for Fedora. Fedora is distributed for free, and RHEL costs money. Fedora is much more cutting-edge than RHEL is, and Fedora has a larger community of users willing to help out and give free support (like this web page).
The "Live Media" is a LiveCD--you can just put the CD into your computer and start the OS without installing it. You can also use the LiveCD to install a very basic version of Fedora, if you want.
The "Install Media" contains all the Fedora packages, and you can install from that without having an Internet connection.
Then there's the choice between i686 and x86_64. Here's how to figure out which one you need:
i686 - If you have a Pentium or Celeron (meaning any Pentium or Celeron, including a Pentium 4, Pentium M, etc.) or the original Core Duo or Core Solo (not a Core 2 Duo or Core 2 Solo). x86_64 - If you have a Core 2 Duo, Core 2 Solo, Core i7, i5, i3, Opteron, Athlon 64, Turion 64, Sempron, or anything newer than these. (This also includes most new "Intel Mac" machines.) Almost every new computer sold in the last few years is an x86_64 machine.
For more details on how to get Fedora, including how to buy a CD (if you can't download one), see the Distribution page on the Fedora Project site. http://fedoraproject.org/wiki/Distribution