Ubuntu Standard Edition: This is the main Ubuntu version, also known as Ubuntu Desktop. All other versions of Ubuntu are based on it. It uses the GNOME desktop environment. Recommended requirements are at least 384 MB of RAM and 4 GB (minimum) of Hard Drive space.
Ubuntu Server Edition: This edition allows you to have a fully working LAMP (Linux, Apache, MySQL, PHP) Web Server up and running in less than an hour. Version 8.04 LTS requires at least 128 MB of RAM and 1 GB Hard Drive space and version 9.10 requires 192 MB of RAM and 1 GB Hard Drive space.
Kubuntu: This is essentially Ubuntu Standard Edition but with the standard GNOME Desktop Environment replaced with KDE and a lot of the standard applications replaced with more KDE-Centric programs.
Xubuntu: Again, this is Ubuntu Standard Edition, just with a different Desktop Environment. Xubuntu is a lightweight version of Ubuntu that uses the XFCE environment. The standard applications on Xubuntu are a 50/50 mix of GNOME and XFCE centric programs. This version is geared towards older hardware.
Lubuntu: A scaled down variant of Ubuntu that is lighter, less resource hungry and more energy-efficient by using lightweight applications and LXDE, The Lightweight X11 Desktop Environment, as its default GUI. Geared towards low-spec hardware that doesn't have enough resources for all the bells and whistles of the "full-featured" mainstream distributions.
Edubuntu: This is Ubuntu Standard Edition with applications geared toward educational purposes. There exists a KDE version and a GNOME version.
Gobuntu: Gobuntu is a completely free (as in speech) Ubuntu derivative. It doesn't have any proprietary software (Drivers etc.) included by default.
Ubuntu Studio: Ubuntu Studio is an unofficial, community-built Ubuntu derivative geared towards Musicians, Digital Artists/Designers, and Movie Makers.
Mythbuntu: Mythbuntu is an unofficial, community-built Xubuntu derivative designed to make building your own DVR (Digital Video Recorder) using the MythTV software suite as easy as possible.
Developers Ubuntu: Developers Ubuntu is an unofficial, community-built Ubuntu derivative designed to make the any kind of development process easier. This is geared towards the developers. It eases the development process by including all the tools necessary for web application development/software development.