The ASVAB is a multiple choice test, administered by the U.S Military Entrance Processing Command, used to determine qualification for enlistment in the U.S armed forces.
Your scores in four critical areas -- Arithmetic Reasoning, Word Knowledge, Paragraph Comprehension and Mathematics Knowledge (see below) -- count towards your Armed Forces Qualifying Test (AFQT) score. The AFQT score determines whether you're qualified to enlist in the military.
Your scores in the other areas of the ASVAB will determine how qualified you are for certain military occupational specialties and Enlistment Bonuses. A high score will improve your chances of getting the specialty/job and signing bonus you want.