Scott:
There is a ton of documentation online about these sorts of problems, so if you want to look it up on your own go for it. However I have a few suggestions that might help.
It seems like your flash plugin for whatever browser you're using has either been corrupted or is missing the appropriate driver for vimeo's coding. (I don't even know if they code differently, to be honest. If you can run youtube videos, but not these ones, it's possible there's a fundamental difference in the way they wrote their video portals.)
What you should try to do is reinstall flash on your browser. Pretty easy step, just click this link:
http://get.adobe.com/flashplayer/I don't know if that helps, hopefully you've already fixed the problem. Also, don't use internet explorer. Firefox (and safari, but mostly just firefox because macs are for trend w*****) is waaaaay better.
- Steve
P.S. if anybody reads this and says 'what an idiot, he should have suggested doing [random tech advice] instead' post up.
Also to you mac fanboys/girls out there, the first step to addiction is admitting you have a problem. Stop sending love letters to Steve Jobs.