More information might help. What vehicle, what year, what engine?

70 Cuda, 340 6-pack motor

Is the solenoid kicking the gear out to the flywheel to engage, or does the starter just free-spin? It is kicking out.
I took the inspection cover off and will look at things later tonight and verify the gear is engaging properly.

Are both wires hooked up? Could your ignition switch be bad?

Wires are connect, new ignition switch relay just installed.

Maybe the "New" starter is defective? Could you take it back and have it checked?

Sure, I could. But it arrears to be working fine when power is applied.

What condition is the battery in?
Battery was low and has been placed on a charger for over 24 hours. I may swap in my Jeep's battery and see what this does.