Refrigerator won't cool
Test the compressor
Your compressor is going to look something like this

To test the windings, you'll first have to remove the &quot;box&quot; where the wires connect.
WARNING: There's likely a capacitor in here. Take care not to let the capacitor discharge into you.

Once you've go that out of the way, you should see three pins/terminals.  They'll typically be arranged in a triangular pattern.

One terminal will be start, one will be run, and the other will be common. To determine if the windings are still good, you'll simply test the resistance between each terminal.

Set your multimeter to read ohms.
Touch one probe to each of the bottom pins.
Write down the reading.
Touch one probe to the lower left pin, and the other to the upper pin.
Write down the reading.
Touch one probe to the lower right pin, and the other to the upper pin.
Write down the reading.

NOTE: Your readings may vary.
You should end up with something like this.

This tells me that the top pin is common, the lower left is start, and the lower right is run. It also tells me the compressor is still good. The reading from start, to run should always be the largest reading. The readings between start and common, and run and common should total up to the reading from start to run.
If the values are way off, or you get infinite resistance between the pins, replace the compressor.
If the compressor checks out, you'll want to make sure it's getting power.
Check the power to the compressor
After reassembling the compressor, plug the refrigerator back in. Set the refrigerator to a setting that should cause the compressor to come on. Using a multimeter set to volts AC (VAC), carefully probe the two wires connected to the compressor.  You should measure ~120 volts.
If you get the proper reading, you'll likely have to replace the relay and/or capacitor. If you don't get ~120 V, the problem is in the control circuitry.
Mechanical failure
If everything checks out, there's still the possibility that the compressor is seized. In which case, you'll have to replace the compressor.