The term "award points" is often used with plans that don't use miles as their unit, such as hotel frequent guest programs and a few airline programs. "Award miles" are used with programs that use miles as their unit, including most airline programs. Either refers to the amount of credit you have in the bank toward their award list.
On AA, there are also "qualifying points" (often abbreviated Q-points) and "qualifying miles" (Q-miles). These determine your eligibility for elite status with AA but have no value for awards. 25,000 of either in a calendar year gets you Gold AAdvantage; 50,000, Platinum; and 100,000, Executive Platinum. (You can also earn the lower two levels by flying enough segments.) You get one Q-mile per mile flown, minimum 500 per leg. Q-points are based on the multiplier in the previous post, but with a minimum of 500 rather than the 250 you would calculate as 0.5x500 for a discount coach fare on a short trip.
A given program or person can use any of these to mean anything that it, he or she wants, confusing matters even further. Who said life was easy?