Basically, Kerbal Space Program is a sandbox (at the moment) spaceflight simulator mixed with Legos (or maybe tinker toys). Basically, you build your rocketship or spaceplane from a list of pre-made rocket parts, put them all into stages, and launch them into space with a -near- perfect simulation of Newtonian physics (minus the Lagrangian points).
The Kerbals are little green men who have recently started up a space agency focused on controlling the far reaches of the cosmos, and seem to "just learn as they go along". The main three astronauts who are the first option to take for space missions are Bill, Jebediah, and Bob Kerman, respectively. Bill is very cautious, but otherwise calm and collected. Bob is apparently extremely frightened of all things space travel (bummer of a career choice) and only seems to calm down when the ship has safely touched down. Jebediah Kerman, the de facto leader of the group, is always smug and happy. If you manage to scare Jeb you know you've done goofed.
If a mod/admin would be so kind as to embed this, this video represents some basic elements of the demo version of the game (and my first attempts at rocket building):
http://www.youtube.com/watch?v=kJM-j8PLW-c
The demo just consists of one command module, a few parts, and the Mun (sic, that's the name of the moon). The full version has its own range of planets to visit, plenty parts and unmanned probe modules, and EVA. Right now I think it's about $18, as it's still in alpha.
Here's a pic of my first attempt at a munar landing (from the demo):
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