STAB stands for Same Type Attack Bonus. When a Pokemon uses a move that matches its type, that attack will get a 50% boost in damage.
So, if I had a Magby and it used Ember, it would get STAB seeing as both are Fire-type. Effectively, Ember is boosted from 40 base power to 60 base power (as 40 * 1.5 = 60).
Dual types like Crobat will get STAB on both their types. In Crobat's case, Poison and Flying moves will get the 50% bonus.
More information on the type chart page.