Have you been getting a loud grinding sound, or just the starter motor going around quickly? If it is the former, the bendix drive on the starter isn't engaging properly. If the starter is just spinning quickly, it is either the bendix drive not engaging at all, or there is a damaged tooth on the ring gear.
If, on the other hand, you turn the key and hear a click, and nothing else, you likely have a defective starter motor. If you turn the key and hear nothing at all, the problem is likely a stuck solenoid.
The Solenoid is the cheapest repair. They are around $20.00, and are mounted on the fender of the car, at the side of the engine compartment. Follow the positive lead from the battery, and there it is. To find the starter motor, follow the second lead from the solenoid to the starter motor. Check the condition of the wires along the way. They could be weak or damaged. They could even be ready to break completely. They are really easy to replace, and the set of 3 is likely to run about $35.00. I always upgrade to 1 gauge wires when I do this.
When you install the new starter, be sure not to over tighten the bolts, or you will strip the threads in the bell housing flange. If this happens, just use a longer bolt, with a double nut on the back side.