The way I learned it, for flats you look at the second flat in the signature from the right. Here, it's A♭ (B♭-E♭-A♭-D♭).
For sharps, take the last sharp and raise it by a half-step. If you had F♯, C♯, and G♯ in the key signature, the last sharp is G♯. Up a half step from G♯ is A.
No sharps is C major, one flat is F major. You absolutely need to know those two by sight.
Note that this is for major keys. For minor, convert however you normally do from major. In this case, the minor key would be F minor.
I highly recommend that you devote some effort to being able to recognise key signatures on sight. You don't want to spend 30 seconds figuring out that you're playing in D major during a sight-reading gig (Actually, this skill sort of comes with practice itself, but certainly don't neglect it).
User Dom found this site, and it has a specific exercise for this. I really like this site, and I think you'll find this useful, especially if you want to memorise them: (The Exercise)
EDIT: If you need help with finding the minor key signatures, just find the major one (A♭), then find its sixth scale degree (A♭-B♭-C-D♭-E♭-F-G), et voilà, you know it's F minor. That, or memorization.