Any format can play on any computer if it has the added codec support. There is no one special universal format. The best modern format for compressed movies is MPEG-4 AVC (known casually as "H.264"). It requires either a modern system or installing free codecs to enable old software (such as Windows Media Player 8-11) to use non-Microsoft formats. Although the H.264 codec grew out of QuickTime MOV container specs (The absolute preferred container format for all professional video editing), it was not developed by Apple and does not belong to Apple. It was developed expressly to have a universal format that had many of the advantages of MOV, but in a compressed form for smaller file size. It does not need QuickTime to play. Starting with QuickTime version 7, Apple made MPEG-4 / H.264 the default format. You would not end up with a MOV file in a modern Mac system, unless you used Final Cut Pro or QuickTime Pro or unless you changed some preference setting for iMovie.
As for a "MOV" container, that plays fine in Windows Media Player 12, as well as in QuickTime for Windows (free if you just download and install it on any Windows 98, XP, Vista, or 7 system). You can use iMovie or QuickTime X to convert MOV to MPEG-4 /H.264 or as MP4 (MPEG-4 Part 14). The former is better. All Vista with Windows Media Player 12 or any Windows 7 system can play modern formats.
You can also use Prism or "Any Video Converter", both free, to convert any MOV to MPEG-4 or MP4. Other converters may work fine, but those two present no noticeable loss of quality.
Although Windows is installed on 87% of home computers in the U.S., Microsoft did not "write the book" on professional video. Apple did. More movies are now played on iPads, iPods, iPhones, and Macs than on PCs.
AVI is an outdated container for video that was not great back in 1992 when it was developed by Microsoft. AVI has no support for b-frames. That means it has limited compression ability. In particular, H.264/AVC codec does not work well with AVI, and H.264/AVC is the new trend for the forseeable future in video encoding. AVI also has no support for attachments, such as additional fonts in subtitles. Since modern formats have none of the limitations and all of the advantages of AVI, we should wean ourselves (this mostly applies to Windows XP users) off AVI ASAP.
The main reason PC geeks hang onto AVI is that 40% of them still use Windows XP, a 10 year out of date system, that is hard-pressed to support modern video. Blame Microsoft for putting a stumbling block in their path to upgrade when Vista bombed. They are just now getting over that awful shock and beginning to make Windows 7 more popular than XP.
EDIT:
AVI does not have problems as long as you don't try to use it with the H.264 codec. IT DOES HAVE LIMITATIONS. Also, AVI is not natively supported by any Apple software, since it is such an outdated format and was never universal (Microsoft only). If you share AVI with users of OS X, they have to download and install special codecs, such as Perian, to enable using AVI with iMovie, iTunes, QuickTime, etc.
Flash video is surely not a good choice for anything except Web presentation, and that only because Adobe made the Flash plugin included with various browsers. It has no useful purpose for non-Web use.
Since all modern Windows systems (Vista with WMP12 plus all Windows 7) support modern format, use MPEG-4 AVC ("H.264") as a universal format.