It is usually expected that all the work included in the demo (code, visual art and music) is originally created by the demo authors (or donated by friends / collaborators). Most of the demo is also expected to be original material, but it is ok to reuse some material from demo to demo (audio samples, textures, models, animations, shaders) as long as it is used in a unique way.

Demos and intros have to run in real time on the target machine without downloading new material on the fly. There used to be fairly generous size limits for the "demo" category, whereas the "intro" categories vary among 64kb, 40kb, 4kb, 1kb and even 256 bytes. They are meant to run autonomously, without needing user interaction. Demo competitions usually include other categories such as standalone music, art of various styles (pixel, 2d, 3d, procedural) and "wild" demos which are the exception in that they are pre-recorded videos.

As with any form of art, there are always people that ignore or challenge some of these rules and assumptions in one way or another, and a few do so in memorable ways, but that's another discussion.