Feel the compressor to see if it is running, it should be vibrating. If it is starting up then shutting down on it's own, I would check to see if it is still getting power. If it is then odds are the compressor is toast. You could check it with a meter, see the link at the end of this post. If you measure a problem then you know that the compressor is toast but if it measures OK it does not eliminate the compressor as it may have internal mechanical problems.

If in fact the unit is still running but is a quiet compressor, I would remove the evaporator cover in the freezer so you can see the coils.
If there is no frost, then either the compressor is toast, you are out of freon or there is a plug in the sealed system.
If there is just frost in one area then you are probably low on freon.
Note: the above checks must be done after the unit has run for several hours, if you check it just after a defrost cycle then there will not be any frost.

In all the above you will need a pro and it will not be cheap.
Also there is no guarantee that they can fix it.
Often it makes more sense to put those dollars into the purchase of a new unit.

OOPS!

Sorry forgot to add the link.
Refrigerator Repair Guide: How To Fix a Refrigerator - ACME HOW TO.com

And here is another good site
http://www.applianceaid.com/frig.html


You will need a meter.
If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.