By any standard, Debian is the most influential Linux distribution ever. Not everyone uses Debian, but, both alone and second hand through Ubuntu, it is the source of more derivative distributions than any other.
How influential is Debian? One indication is that three of the four most frequently downloaded distributions on Distrowatch are based on Debian: Ubuntu, Mint, and Debian itself. Together, these three account for 49% of the top ten downloads. Assuming that these downloads are representative of the interest in Linux, just under half of anybody's experience with the operating system comes through Debian or a distro based on Debian.
Or, to give a different metric, of the 323 currently active distributions listed on Distrowatch, 128 are based on Debian, and another 74 on Ubuntu. In other words, just under 63% of all distributions now being developed come ultimately from Debian. By comparison, 50 (15%) are based on Fedora or Red Hat, 28 (9%) on Slackware, and 12 (4%) on Gentoo.
Given these figures, it is not surprising that Debian should have its Derivatives Front Desk and Ubuntu its Derivatives page to track their relationships with other distributions. As incomplete as both these efforts at keeping track currently are, they are still additional proof (if any is needed) of Debian's and Ubuntu's far-reaching influences.
Start scanning the lists of derivatives, and you will find something for everybody, from general purpose deskstops, Live Media, alternative interfaces, netbooks and other platforms to compact installations, localizations, security and privacy, and multimedia.
Debian was the last of the major distributions to get a user-friendly installation. To fill this gap, many derivatives sprung up, including Libranet, Stormix, Progeny, Linspire and Corel, all of which became defunct long ago, due to major changes in the Debian Installer and to Ubuntu's basic installer.
Today, the main survivors of these distros are the commercial Xandros, which began with the Corel code and later purchased Linspire's assets, and MEPIS, a KDE-centered distro that continues to advertise itself as simple and easy to use.
These survivors are joined by Linux Mint. Linux Mint's main advantage over Debian and Ubuntu is that it installs proprietary enhancements such as Adobe Flash as part of its standard installation. Available in both Debian and Ubuntu editions, Linux Mint also offers a wide variety of desktops, ranging from the standard GNOME and KDE to lighter weight alternatives such as Fluxbox, Xfce, and LXDE.
Another all-round distribution is gNewSense. Its main claim to fame is that it is a completely free distribution that carries no proprietary software whatsoever. In fact, of all the free distributions recognised by the Free Software Foundation, it is by far the most polished and usable.
One of the oldest derivatives is Knoppix, which is also a pioneer of Live Media -- CDs or DVDs from which you can boot a computer without accessing the hard drive for secure computing or for demoing software. For many, Knoppix remains an essential tool, especially as a rescue disk.
In fact, Knoppix is so popular that it has evolved its own derivatives. These derivatives include kademar and KnoSciences, both of which have more extensive hardware support than Knoppix, the clearly named STD (Security Tools Distribution, and VMKnoppix, a sampler of virtualization alternatives. Like Knoppix itself, these third-generation derivatives can be used from an external drive or installed on a hard drive.
Debian has a long tradition of including alternative desktops and window managers in its repositories. Similarly, Ubuntu includes a class of what it calls "recognized derivatives" that it considers part of the project, and which includes Kubuntu (KDE) and Xubuntu (Xfce).
However, these are by no means the only interface alternatives. If you install Bodhi, you can work in Enlightenment, the window manager that borders on being a desktop. Alternatively, Damn Small Linux and the dormant Fluxbuntu both use Fluxbox as a window manager. If Fluxbox doesn't fit your needs, then perhaps CrunchBang and Openbox might instead. Still another derivative that has received considerable attention in the past year is Lubuntu, which uses the lightweight LXDE desktop.
Although they have not been universally popular, developing a desktop for netbooks has been a main concern in free software for the last few years. Of course, Ubuntu itself is developing the GNOME-based Unity for its main interface, a project that was originally called Ubuntu Netbook Edition, but you can find no shortage of netbook-centered derivatives, either.
Details of the interfaces differ, but many of these netbook designs feature a simplified desktop, often with the main menu replaced by large buttons on the desktop, or with panels down the left side of the screen, moved there to take advantage of the extra horizontal space on wide screens.
The history of Debian and Ubuntu is riddled with derivatives that briefly attempted ports to other hardware platforms before being discontinued. These include BlackRhino, which ran on Sony Playstation 2, and a number of derivatives designed to run on Openmoko's Neo Freerunner.