A better option is to start your own Youtube account and host them there then link back to them from your post. You can either embed a Youtube video directly into the post and slow the forum down or just link to it so that it opens separately and allows people to keep browsing the site if the video takes time to load.
With other peoples vids just link to them using the chain link icon in the "reply box'
Edit: Or do you mean take someone elses video and store it on your computer? If so yes it is possible but it would be far nicer to ask the person before doing it especially if they are a member of the forum. If they aren't a member of the forum steal it off YouTube.
Technically stealing is downloading and sharing is uploading but so many people confuse the two so I gave you options.
