Well happy birthday!
'C' stands for "Common". Both the compressor and capacitor have terminals marked this way. In the compressor the two windings (S - start and R - run) are connected together at one end and that connection goes to the common terminal. The free ends go to, respectively, the R and S terminals. In the capacitor can one terminal from each of the two individual capacitors in the can are connected together and to the common terminal. The other sides of the capacitors go to, respectively, the FAN and HERM terminals.
There are, apparently, three black wires going to the capacitor. Two are from a start assist which appears to be under the transformer in the photo. The two wires from that go to the capacitor C terminal and the other to the capacitor HERM terminal. The third black wire comes from the R (run) winding on the compressor. It also connects to the C terminal on the capacitor. This is the one that goes to the "black round thing up top" which I am assuming is the thing to which the red, black and blue wires all run together, the thing with S, C and R molded into the top, the left of two cylinders on top of the compressor. That's not the thermostat. That's where the wires enter the compressor through a glass hermetic seal.
In summary, there are 3 terminals on the capacitor. One is labeled "FAN" and should have a single wire of color "TM", whatever that means - it appears to be yellow in the photo. The second is labeled "C" and should have a white wire running from it to the fan motor, a black wire running to the R terminal on top of the compressor and another black wire running to the circuit board start assist. The third terminal is labeled "HERM" and should have a single red wire running from it to the S terminal on the top of the compressor.