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 the watchlist display by clicking "watchlist" link in the top right, however this is only available to users who have registered and logged in.
Before you see anything interesting here, you must add some pages to your watchlist. Do this by navigating to a page which interests you, and clicking "watch" in the page tabs (located in drop-down menu).
The watchlist feature is mainly useful when The Vault - Fallout Wiki becomes very busy and the recent changes display becomes too high traffic. During a low traffic period, you can normally see all of the editing activity across the whole of The Vault - Fallout Wiki without feeling overwhelmed. On a more busy period 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 chose to watch the page, to see if there are any follow-up edits for example. Note that there is a "watch this page" tickbox provided on the editing screen 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.