YouTube has multiple servers for serving images of the videos, hosted on the site. They are mainly used for load ballancing (so you get the image that is hosted on a server, that is physically closest to you so it loads faster).
However, as a developer or a website owner you may want to programmatically build a link to the thumbnail or still image that YouTube uses for a given video.
What you need is – the ID of the video (e.g. if the video in question is http://www.youtube.com/watch?v=m3JAtvsqfF8, the ID would be m3JAtvsqfF8). When you have the ID of the video, just replace it in the below URL and open it in your browser.
http://img.youtube.com/vi/video_id_here/0.jpg
You should immediately see the full-size thumbnail that YouTube uses across its website (note that scaled down versions are also available; just replace the last zero in the URL with “1.jpg”, “2.jpg” or “hqdefault.jpg”).