Very simply, Simple DirectMedia Layer, or SDL for short, is a library that allows you low level access to a video framebuffer, audio output, mouse, keyboard, and joysticks across a wide variety of operating systems.
The current version supports Windows XP and newer, MacOS X 10.5+, Linux 2.6+, iOS 5.1.1+, and Android 2.3.3+. SDL includes code for other operating systems but they are not officially supported. See the page about Installation for the most recent list.
The latest stable version is 2.0. You can always get the very latest bleeding edge Git snapshot, which may be broken at any given time, from the GitHub project page.
You can send questions and feedback through the SDL forums/mailing list or using the Feedback link (bottom of the page).
You can submit bug reports to SDL's GitHub issue tracker and, if you like, send patches as GitHub pull requests.
We run a Discourse server here, which can be put in "Mailing List Mode," where you create an account on the website, flip a switch in the preferences, and then can interact exclusively through email thereafter.
Yes, you can find people hanging out on the IRC server at irc.freenode.net on the channel #SDL
Yes, go here!
Yes, the official SDL logo was created by Arne Claus, and is available here: http://www.libsdl.de/logo/ Several community contributed logo ideas are also available at that URL. You are welcome to use any of them, as well as any artwork on the SDL website, to represent Simple DirectMedia Layer.