Mike wrote:
Last evening, my A/C unit stopped working. I'd like to try to figure
out what died, how much it will cost to fix, and if I can do the
repairs myself.

We heard a sort of buzzing sound a few times, and when I looked out
the window, I noticed that the fan blades seemed to be turning slower
than usual. I went to the thermostat and turned off the A/C for a few
minutes. When I turned it back on, I could hear a short buzz sound at
the A/C unit, but the fan blades didn't move. The buzz sound would
repeat every 10 seconds or so, but the fan never moved.

I then turned off the A/C and flipped the breaker. I tried to spin the
fan blades with a screwdriver, and they turn freely. I checked the
service disconnect box outside the house, and all looks fine.

I'm thinking that either the fan motor died, or the compressor is
dead, or both. Any thoughts?

Probably neither one. A seized motor sort-of "hums." The only thing that can
"buzz" (like a bee) is the power relay.

You can try this: With all the power off, look at the contacts on the power
relay. The contacts may be pitted or burnt, thereby preventing a good
connection. If so, more life can sometimes be extracted from the relay by
filing the contacts down to a smoother surface. This may buy you some time
to get a replacement.

While you're at it, might as well replace the capacitors.