The only general requirement is that there is no general requirement. Most contests will do things differently, and it really helps to be prepared for whatever you may have to do, from sending in an insane bitrate MPEG2 encode to submitting on Beta. OK, so Beta isn't likely a format you'll have to worry about. But it does really help to be flexible. The means first and foremost: Never delete your 'master copy'. Always keep around (even if that means archiving it to CD/DVD whatever) a copy of your highest quality copy of your video. That way if you want to submit it to a contest that has strange requirements that you don't already have an encode of it for, you can make the best quality copy possible.
In general these days, most cons seem to be moving towards accepting files digitally, either on mailed-in media (CDs,DVDs) or over the internet (FTP) in high-quality MPEG2 format. For a bit of an idea of what I'm talking about, look at the AWA VAT rules website (
http://www.awa-con.com/lvat.htm)- it has a fairly good description of the encoding that they want, which is a good starting point. Many places will accept authored (playable in a regular DVD player) DVDs, so if you have a DVD burner, that is always an option. It's also good to have some method of getting your video onto a VHS tape, as there are some conventions that still only take submissions on tape (though they are getting fewer and more distant between).
In short, read up on the contest(s) you want to enter, and go from there. And never delete your master video file!