The main two differences between Ubuntu and its derivatives are the backing and the default installed packages. For all official derivatives, you can convert between variants by installing certain packages.
Here is a small list:
Ubuntu comes with the Unity desktop environment. The underlying Unity platform is still GNOME, but instead of using the GNOME Shell interface, Unity uses the Unity shell.
Ubuntu and Unity are commercially backed and supported by Canonical.
To convert an installation into regular Ubuntu, install ubuntu-desktop 
.
This is the KDE flavor of the typical Ubuntu (which is GNOME driven). The primary difference is that Kubuntu comes with KDE as the default Desktop Environment, as opposed to GNOME with the Unity shell.
Kubuntu is sponsored by Blue Systems. Canonical stopped backing it in 2012, but it is still an officially recognized Ubuntu variant, which means that it gets build machines, test machines, CD image distribution servers, etc. dedicated to it.
To convert an installation into Kubuntu, install kubuntu-desktop .
Ubuntu GNOME aims to have a mostly pure GNOME desktop installed by default.
Ubuntu GNOME is community-driven. It was originally called Ubuntu GNOME Remix; it was renamed to Ubuntu GNOME when it became officially recognized. Ubuntu GNOME is very new: its first release as an official derivative was 13.04 Raring Ringtail, which as of 6/9/13 is the current release.
To convert an installation into Ubuntu GNOME, install ubuntu-gnome-desktop .
This is a lighter weight, highly efficient and optimized flavor of Ubuntu designed to run on older computers. It uses Xfce which is a proven faster Desktop Environment than both KDE and GNOME. This is a typically simpler slim interface.
Xubuntu is community-driven and operates under a Strategy Document. It is also officially recognized by Canonical (see Kubuntu for what that means).
To convert an installation into Xubuntu, install xubuntu-desktop .
Ubuntu Server is optimized for use as a server. It does not come with X.Org, and as such does not use any graphical environment like a desktop environment or a window manager. Instead, it comes only with a CLI environment.
Ubuntu Server is commercially backed and supported by Canonical.
The counterpart to Xubuntu - Mythbuntu is designed to be an entertainment powerhouse. Focused around being a Media Center it includes many drivers and setups for TV Tuners, TV Out cards, and has a Media Center application (MythTV) integrated into the Desktop Environment to facilitate the entertainment powerhouse it advertises.
Mythbuntu is an officially recognized Ubuntu variant.
To convert an installation into Mythbuntu, install mythbuntu-desktop .
Lubuntu takes the aims of Xubuntu and pushes the Desktop Environment to an even more bare bones lightweight variant:
- LXDE (Lightweight X11 Desktop Environment) in releases up to 18.04, and
- LXQt in releases after that.
Both are very efficient (LXDE used GTK2, it was being ported to GTK3 where it was deemed to heavy, so the devs ported it to Qt5, they then joined the Razor-Qt team forming the newer LXQt project). It's a more efficient (the lightest official flavor out of the box), power saving, fast, lightweight Desktop Manager than XFCE (though if using GTK3 apps some of its lightness will be lost).
Lubuntu is community-driven. It is officially recognized as a variant by Canonical.
To convert an installation into Lubuntu, install lubuntu-desktop
My Opinion
I've used the first three before and I believe all have some backing from Canonical (or other corporate backing) while Lubuntu is still relatively new and community driven. Though if you have a very old/slow computer it certainly might be what you're looking for.
Figured I'd add the other flavors of Ubuntu I knew where out there
In addition to the above listed there is also:
This flavor is designed with Educational intent. Runs very close to the vanilla Ubuntu release though it comes with many additional applications that are best suited for an Educational environment. It also is configured and stylized with kids in mind.
Edubuntu is an officially recognized variant.
This flavor is geared towards those who deal with multimedia (Video, Audio, Graphics, Design) on a daily basis. Comes bundled with many applications, codecs, and drivers required to facilitate those activities.
Ubuntu Studio is an officially recognized variant.
Ubuntu CE (Ubuntu Christian Edition) is Ubuntu designed for Christians who wish to maintain an "Christianly" lifestyle. This comes bundled with a lot of religious software and security tools to help protect moral religious obligations.
Ubuntu CE is an unofficial derivative.
This project is idle, but it's intent was to bring stronger hardened security to the stock Ubuntu installation. It targets security practitioners like penetration testers.
Nubuntu is an unofficial derivative.
Fluxbuntu does not come with a desktop environment like GNOME or KDE. Instead, it only comes with a window manager called Fluxbox.
Fluxbuntu is community-driven. It is an unofficial variant.
Best Answer
Regarding the switch to Mir, please read the latest update of the Ubuntu Graphics Stack Roadmap. In short, Mir will be default compositor in 13.10, yet everything will still use x11 through xmir an x.org will be available as fallback. In 14.04 x.org will be completely removed and everything should run through xmir. Starting from 14.10, Mir itself will be used by Unity, and xmir will only be there for backwards compatibility.
Debian, on which Ubuntu is based, will not switch to Wayland or Mir in the foreseeable future, so I expect x.org to be available in the Ubuntu repositories still for quite some time.
According to a post in the phoronix forums, no other flavour of Ubuntu will use Mir anytime soon and probably none of them ever will. Those that rely on toolkits ported to Wayland will maybe switch to Wayland.
(Sidenote: If you have time, read also this slightly *cough* biased posting, it lists possible reasons.)