Why isn't there a server edition for Fedora like Ubuntu?
Fedora only has a Desktop editionhas Server edition. In my experience, "server" distros have a longer support lifetime than desktop distros. If you want something that has a longer support cycle, use Red Hat Enterprise Linux, or one of the RHEL clones like CentOS. Fedora really isn't suited for a long-term support OS, because at best you'll get 13 months of support if you install a release the day it comes out (Unlike RHEL).
While Fedora is really only aimed at the desktop, the OS certainly supports being installed as a server, using the DVD or network installer. RHEL6 is based on Fedora 12 (with backports from later Fedoras and security fixes) and uses the same installation software as Fedora, so there's nothing stopping you from using Fedora as a server OS, just don't expect to get updates or security fixes past the end of its support lifetime.
RedHat Enterprise Linux is Fedora server.
Fedora is where RedHat innovates and so it's bleeding edge by definition and not meant for servers. RedHat Enterprise is based on Fedora. Fedora provides a platform to develop and test new updates before they get pushed to RedHat Enterprise.
This is why Fedora gets the newest stuff before other distributions. This makes Fedora a great desktop OS for early adopters who like to get the new stuff first, but for servers, you want battle tested code, not bleeding edge.
Fedora is such an asset to RedHat because by the time something makes it into RedHat Enterprise, the code has been hardened after being battle tested in the wild by millions of users.
RedHat has been in the game for a long time pushing Linux development, and the Fedora model has proven effective. RedHat's first release was in 1995, Fedora was released in 2003, and in 2012 RedHat became the first billion dollar open-source company.
In essence, Fedora is RedHat Linux before the addition of RedHat Enterprise Linux. The name was changed to reduce confusion, but the project's importance was made clear when they renamed it after RedHat's iconic symbol -- its red fedora.
See the Red Hat/Fedora project announcement.
Running Fedora on your desktop keeps you at the forefront of Linux development world (it's what Linus uses), and it makes your life easier if you're running RedHat or CentOS servers because the environments are the same. In contrast, if you run RedHat on your servers and Ubuntu on your desktop, you have different package managers, file system layout, etc.