For those who might not know and have custom firmware installed in their PSP, there is a homebrew application called PSP Tube. Which is really great for downloading videos from YouTube and other video sites. All you need after you install this application is a good memory stick that will hold the space needed for the downloads and an internet connection via Wi-fi.
If you do not have a PSP with custom firmware then your best option is to convert the videos so the PSP will be able to play them. PSP only plays MP4s and I think WMA files, (there might be others I do not know just yet) but MP4s are best. I downloaded this software called "PSP Video 9" it is absolutely free software. You can click on the link and download it. It is easy to use and I recommend it to all. So have fun watching your favorite video on your PSP.