what is this relay called?

I think you are being deceived. Either what you hear is not the problem, or you are hearing the starter relay out on the firewall clicking, and there is something wrong in that circuit, either the starter relay is going bad, or the starter.

Is it just one click when you hold the key to start, or a buzz, rapid clicking when you hold the key to start
With the key in on position and tugging (to the left) on the trans selector towards park, I hear one click and I can touch the relay behind the dash and feel it click as well. I then turn the key to start the car and the engine starts. If I don't hear/feel this relay click (the only way to achieve this is to move the trans selector more towards the left in the park position) I get absolutely nothing from turning the key to the start position....