The thermostat is set at 72 degrees. The unit's fan begins turning within a few seconds there is a buzz then a click. The fan continues to turn properly but every few seconds there is a buzz. The unit is not cooling the house properly. What could be the problem?

The buzz and click is most likely the compressor trying to start, then cutting out. Could be a bad compressor capacitor, or a weak compressor in need of a hard start kit, or a bad compressor. I would start by having the compressor capacitor checked, and if good, add a hard start. This of course after checking for correct voltages to the compressor. Lots of options here, but Im betting on a bad capacitor (which is actually good news because capacitors or even hard starts kits are fairly cheap in comparison to the cost of a compressor or complete condenser.