Is Fedora the free version of Red Hat ?

If not, what exactly is Fedora ?

Well, "RedHat" is a company. There used to be distributions called "RedHat", the last one was "RedHat 9". Then they stopped doing them and instead started producing Fedora Core distributions, versions 1-6. After that they got a new idea and renamed the series to just "Fedora" (from 7 onwards). Those aside there are then the commercial RedHat Enterprise things, whatever they're preferred to call then..RHEL is the name you see everywhere ("RedHat Enterprise Linux"). As I understand it, and please correct me if I'm talking nonsense, Fedora is not the same as those commercial RedHat Linuxes, not anymore at least. At some point (during Fedora Core series) it was said that Fedora Core was a sort of "testing ground" for the commercial RedHats, that software was released in Fedora Core, bugs found, patched and after that they took the software to the commercial RH release; the versions indeed did go almost hand in hand in the past, so that Fedora Core was one version ahead of the commercial one. Though I doubt if that's the case anymore.

Anyway Fedora is an "open" distribution, and I take it that it has it's own development teams and such, and is not that strictly connected to the commercial RedHat series. They surely share something, but most Linux distributions do anyway. If Fedora or RedHat site has something to say about this, it's there and you can read it, and if it doesn't, there's not much room to wrestle about it. I hope the commercial RedHats are more stable than Fedoras, and probably are, because I assume (or hope) that the commercial ones use not-so-bleeding-edge software than Fedora uses, and thus would have less bugs left in them to annoy the users.

Anyway both of them, commercial and non-commercial RedHat products are basically "the same" system, except that you don't get paid-for support for Fedora (unless you buy it, of course). Commercial RedHats might have some sort of system upgrading tool of their own, that only works after you've "registered" or something, but these are just my guesses - the point is, what you can do with (commercial) RedHat, you can do with Fedora. It's just that industry likes to pay for support, and it's not a stupid idea most of the time..

Oh, and don't forget the commercial RedHat derived distributions. I recall there was a distribution that resembled "RedHat" and had a word in it's name that said something about "white", and was basically a commercial RedHat 'ported non-commercial' if it can be expressed in that way. But it's not the only one as far as I know, and doesn't need to be - the commercial RedHats use opensource code like Fedoras do, and that means it's open for the rest of us too.

The free RedHat Enterprise Linux Clones (built from the RHEL sources) are:

CentOS
Whitebox
TaoLinux - Dead project

personally if you are going this route I would suggest CentOS as it is the most current and well supported of the clones from what I have seen. http://www.centos.org/

Waht b0uncer said is in fact the case, also farslayer.
Just to add a bit, fundamentally, Fedora is more bleeding edge and high turnover of new editions (every 6 -12 mths ?).
RHEL is much slower tunrover, commercial support is/updates are good for 5 yrs (last time I looked).
RHEL is more stable, not leading edge ie this is what high powered industry wants, stable/reliable/paid for support.
I'm fairly sure RH does not supply any (paid for) support for Fedora, although the Fedora proj wbsite has the usual options eg Docs/FAQ, and of course here we have spearate sub-forums for RHEL & Fedora.
As farslayer said, the most popular free version of RHEL is Centos.

HTH