Electrically, here's what the circuit looks like. I've removed the ST SIGN circuit for clarity.
Ben's right. This sounds like classic solenoid. On the image, the starter and solenoid are represented as one, so I drew a green line to seperate them.
The solenoid does two things:
1- It extends the starter gear into position to mesh with the flywheel, and
2- It closes an internal switch contact to send power from the big black wire to the main starter motor. (Essentially a relay)
It sounds like the gear is extending, but the switch isn't closing.
When you crank the starter, does the voltage in the car drop to 0? (ie: do the panel lights go completely and totally dark?) If not, then your battery is fine.
(edit) actually, that green line should be one coil further right. Oh well, you get the idea.