The Demo version of the game is basically a snap shot of the game as it was up to version 0.13.3. There is nothing significant about the development of the demo version, no special side branch or purposely locked feature, it is simply the 'latest' version of the game at that time. It is available to play for free and provides a very good demo of how the sandbox mode (currently the only game mode) works in Kerbal Space Program; it does however offer a much reduced number of parts and celestial bodies.
The Star System
The star system that is featured in the demo version features only the star Kerbol, the main planet Kerbin and its one initial moon Mun. The physics model has been improved since the demo version, so ships designed in the demo version will handle very differently in the latest paid version, usually a lot worse. That said, the stats for the parts in the demo version are tuned more towards the physics model in use at the time, so the game is by no means broken in the demo version.
As mentioned, the parts available in the demo version is a lot less then in the paid game Template:Demo Parts
How does this wiki refer to the demo version
This is more of a side note for wiki developers, but for those reading will help explain why things are done they way they are. Initially, parts where given a page for the paid version and a page for the demo version; this decision has since been regarded as foolish, as the parts are still the same, and in terms of functionality are exactly the same.
Instead, as agreed by N3X15, parts will simply receive one page for both paid and demo version, with notes made where applicable about how the demo version of parts differ from the current version. The difference will usually just be a case of differing stats for the part.