When I turn on the unit, I hear the unit engage. I set the set pt very low for cooling. The condenser on the outside comes on and the copper pipe gets cold. Sounds like the blower fan is on, but I hear a constant clicking switch noise from the unit. Is it the thermostat, or a relay?

Not sure what you are talking about.. more info please.

Yes, more info. Is the condenser fan running? Is the blower inside the air handler? Where is the clicking coming from: condenser or air handler?

The fan on the outside actually comes on. The clicking is in the air handler unit in the basement. The blower is also in the air handler unit in the basement. Although the system is set to cooling, the unit just produces heat.

The fan on the outside actually comes on. The clicking is in the air handler unit in the basement. The blower is also in the air handler unit in the basement. Although the system is set to cooling, the unit just produces heat.

What is your backup/auxiliary heat source: furnace or heat strips? Does the air coming out of the vents feel like room air temp, or much warmer like the heat is on?

You could have a bad relay, reversing valve, 24 volt problem, tstat, or emergancy heat/secondary heat problem.
Also your compressor might not be coming on.

Honestly I suggest having a tech come look at it.

I agree with sibair. Make sure you get a tech that is very knowledgeable with heatpumps, they can be a real pain. Good luck and please let us know how you make out.