Yes, I know my engine bay is filthy; go suck an egg.
If you'll notice the accumulator in the center of the pic (black canister) you'll see a plug with two wires going to it, a purple w/white and a blue w/yellow (IIRC). This plug is the connection for the pressure safety switch. It senses pressure and if it drops too high or low, it will kill power to the compressor clutch (by opening the circuit).
In order to verify that this is the culprit, pull that plug off and run a jumper wire between the two terminals inside of it (the plug, not the actual switch on the accumulator). Now turn the AC back on. It should come on and stay on. Now shut the AC off.
If it did indeed come on and stay on, the pressure safety switch is either working or it isn't. What this means is you either need the switch replaced or you actually do have too high of pressure in the system. Either way, based on your self described lack of AC knowledge, you need to take it to a shop.
After completing this test, be sure to plug the wiring back in. DO NOT, UNDER ANY CIRCUMSTANCES LEAVE THE SWITCH JUMPERED AND RUN THE AC!!! If there is actually a high pressure situation in there, you could easily destroy the compressor by over working it. Also, when doing this test, run the AC with the switch jumpered for as little time as possible. You just want to confirm that it isn't starting and stopping like you described earlier and that it is a condition of the switch causing it..