I think the FAQ, under the "Can I search the forum?" topic, should contain all the gory details about searches, whether full boolean expressions are allowed, what, if any wildcards are allowed (they are, BTW), how the quotes work, lots of things.
If you search for "search illegal" in quotes, it will only look for that exact phrase. It won't find "illegal search" unless you search for that exact phrase, in quotes. If you search for "search" and "illegal", both words, no quotes, it will find all posts with either the word search or illegal in it.
If you do a search without quotes, "search AND illegal" with the 'AND' in capital letters, it will find all posts that contain both the words search and illegal, just not necessarily that specific phrase.
Same thing with 'OR' instead of 'AND' in that it will find all posts that contain either search OR illegal.
'Illegal NOT search" without quotes will return posts that contain the word "illegal" but do NOT contain the word ""search".
That should cover the Boolean basics, anyway. The drop-down menus lets you further refine your search, like only from a specific user, only in specific forums, returned as threads or posts, etc.