The PSP is supposed to play Flash video one of these days ... but considering Sony's slow adoption of features we really want, we wouldn't hold our breath for it. Until then, there's a ridiculously easy solution for converting videos from sites like YouTube ... and it doesn't even involve downloading any software!
1. Get the URL of the video you want.
2. Copy URL into the "FLV Online Converter" at vixy.net
3. Choose "MP4 for iPod/PSP"
4. Wait for the conversion, download file, and then save it to your PSP under the "VIDEO" folder.
Does it get any easier than this? Not really. I found it to work quite quickly, with some impressive results.