An AVI, as it applies to Chip's Challenge, is a term for a video solution to a level captured in .avi format, which stands for Audio Video Interleave. In the older days of CC, SnagIt:  was generally used to make AVIs, and it recorded only at 5 frames per second. This caused many routes to be jerky, especially heavy boosting routes. Many of the older AVIs have been re-recorded by Andrew Bennett, with minor tenth improvements added to them as well.
Eventually, ChipCap was written by Anders Kaseorg for the specific purpose of making AVIs. Hypercam also works well for recording videos. Generally, when discussing their scores, Chipsters will have written an AVI for a level if they wish to share their route; some players, however, prefer to challenge others by not revealing their solution.
Nowadays, Tile World's automatic TWS playback feature is also used to share solutions, specifically as the "public TWS" at David Stolp's website. The public TWS automatically updates itself when a person loads a higher-timed route, even by only one-tenth on one level. To download the solution files to CC1 and CCLP2's levels, go to the site, and on the bottom are links to the pages for both sets, plus one for a famous level set by Evan Dummit, with instructions on how to use them written in the pages. However, Tile World 1 requires the user to play the entire solution out, rather than fast-forward to a specific part.
The ChipWiki Youtube channel contains the AVIs for all 149 CC1 and CCLP2 levels. The level walkthroughs use these, with specialized annotations to explain the routes, especially difficult routes (generally involving boosting, Slide Delay, routes which normally work best with sound, or many pauses) that would be hard for the average viewer to decipher from simply watching them. This project is currently in progress and is handled mostly by Andrew.