An ABA law school allows you to take the Bar examination in any of the 50 states. A non-ABA law school only permits you to take the Bar examination in the state where the school is located and you cannot practice law outside of the state or take the Bar examination for different state.
Without ABA accredidation, graduates from a particular college may not be allowed to practice law in many states. ABA accredidation is important because it gives the school recognition and proves that meets all requirements to produce graduates ready to practice law. Students at ABA accredited schools can join the ABA at reduced rates instead of only having associate membership.
An ABA accredited law achool has undergone a rigorous self-study and accreditation visit through the American Bar Association. This is a guarantee that the school is dedicated to a high level of competence in instruction, and that the faculty are highly qualified. A non-ABA accredited school may also be good, but doesn't have the same guarantee.