OK now I get it.
I did a Google on it and found some info.
It checks for AC voltage in a wire.
It does not tell you if there is current flow just that there is AC on the line.

It is probably just detecting the electromagnetic radiation given off by 60 hertz AC.
It does not tell us what that voltage level is.
Also I am not familiar with it so cannot say if it would pickup radiation from nearby wires and give a positive indication.

Anyways if the start device is getting voltage and the compressor does not start then it is either the start relay or the compressor itself.

FYI: If you use a water hose/pump set up as an example:
Voltage is the amount force created by the pump to push the water through the hose.
Current is the amount of water being pushed though the hose.
Resistance is any resistance to the flow of water through the hose. As an example if you step on the hose.

[COLOR="Blue"]The compressor runs for about 20 seconds and stops, though. Is that normal? [/COLOR]
No and it is really bad news as it sounds like the compressor is toast.
[COLOR="Blue"]
Also, the freezer seems somewhat colder, but the refrigerator is about the same. Any ideas, or am I too impatient?[/COLOR]
Yes even if the compressor was running properly you would be impatient.
A fridge will take at least 12 hours to get to a stable temperature.

Any idea how long the warranty is for a compressor? Also, is an actual warranty necessary, or will the original receipt (have mine in-hand) for the purchase suffice?
ct