The contactor.

The contactor is an electrically operated switch in the outdoor unit. The thermostat sends it a low voltage signal, a coil inside pulls an armature down, and two sets of contacts close to feed the compressor and the fan their line voltage.

It is the part that makes the audible clunk when the outdoor unit kicks on.

How it wears out

Every time those contacts close they arc a little. Over thousands of cycles the surfaces pit and carbonise. Eventually they stop making clean contact, and the result is a unit that will not start, starts intermittently, or chatters instead of pulling in cleanly.

A contactor that is welded shut fails the other way: the outdoor unit runs continuously, even when the thermostat is satisfied, because the switch physically cannot open.

The Arizona version

Ants are drawn to the magnetic field around the coil and get in through the bottom of the cabinet. Enough of them in the wrong place will hold the contacts apart or short them. It is common enough here that it is worth looking for rather than assuming the part simply aged out.

What it costs

It is a flat price on the list, and it is one of the cheaper real repairs on a system. Where the fault is not obviously the contactor it starts as a diagnostic, and you hear the repair number before anything is touched.

If you have been quoted a figure with a comma in it for a contactor, that is worth a second set of eyes.