Your watchlist is a display for tracking changes, similar to the Recent changes display, but limited to showing changes only across your set of "watched" pages. You can see your watchlist by following the "Watchlist" link in your account menu (on the top right). This feature is only available to users who have registered and logged in.
Before you see anything interesting here, you have actually add some pages to your watchlist. Do so by navigating to a page that interests you, and click the star icon (File:Vector skin - page not in the watchlist.png) or "Watch" tab.
The watchlist feature is mainly useful on a very busy wiki where the "Recent changes" display becomes too overwhelming with many changes you don't want to follow. On a typical low traffic wiki, you can normally see all of the editing activity across the whole wiki without feeling overwhelmed. On a more busy wiki you will find that many edits are taking place on pages which you personally find less interesting. To help track changes which are more interesting, start watching pages!
Typically you would watch pages related to your interest areas, and also pages where your care most about the content. If you make a significant edit to a page, you might want to watch the page (so you know when there are any follow-up edits). Note that there is a also a checkbox on the edit screen ("Watch this page") for this purpose.
On the watchlist display, some page names appear in bold. This indicates whether or not you have visited the page to see the change. It's a bit like read/unread emails in an email inbox. The "Recent changes" display will also have some new bold highlighting, but for a different reason. Here bold is just indicating edits to pages on your watchlist.