To post a video on a forum, you usually put the URL (web address) between two tags indicating it is a video. Those tags will vary depending on the type of forum software and the video source.
On this messageboard, if you hit "new reply" you will see options for formatting your text and inserting various elements. The video option is indicated by an icon that looks like a television. There is a drop-down menu next to it that includes seven common sources of videos.
You simply select the source and paste the URL for the video into the box that pops up and press "OK."
There is a forum I frequent that doesn't use a drop-down menu. When I want to post a video, I go to a post that includes a video and hit reply (with quote). I then copy the code for the video in that post and start a new post. I paste the code I copied into the new post and then change the unique identifier.
For example, from this:
http://www.youtube.com/watch?v=b1ed3bSjqSI
I would paste this: b1ed3bSjqSI over the existing identifier.