My uncle gave me a computer that he says doesn't work, and told me that if I could fix it, I could keep it.
CPU station is connected with the power plug, and it's turned on. Attached is a monitor for another computer I use, and it's turned on and plugged in. (So the monitor is guaranteed to be working, since it's not my uncle's monitor.) All wires, cables, and parts from inside his CPU are in their proper place, none have been tampered with or removed. No other devices are plugged into the CPU.
The processor fan is spinning, the power unit fan is not spinning, the monitor shows a black screen after turning on. IE: It is receiving no apparent display signal from the computer, even though it is on.
What should I check for now, or is it the power supply unit that is damaged since its fan isn't spinning?
Full specs, as best as you can give, would help. Also, how much do you know about computer hardware? Is this a home built computer or something like a Dell or HP?
Having said that, your symptoms (PSU fan not turning on) would seem to indicate a PSU problem. If your specifications show a crappy power supply, that increases the probability that it is the PSU.
The rest of my post includes a slightly edited version of my "standard" troubleshooting procedures.
Keep detailed notes of what and where you unplug and plug in cables and stuff. I have been building computers for years and I have decades (about 4 of them ) of experience maintaining complex military electronics systems, and I still keep notes when working on my computers.
Try to verify (as well as you can) that the PSU works. If you have a multimeter, you can do a rough checkout of a PSU using the "paper clip trick". You plug the bare PSU into the wall. Insert a paper clip into the green wire pin and one of the black wire pins beside it. That's how the case power switch works. It applies a ground to the green wire. Turn on the PSU and the fan should spin up. If it doesn't, the PSU is dead.
If you have a multimeter, you can check all the outputs. Yellow wires should be 12 volts, red 5 volts, orange 3.3 volts, blue wire -12 volts, purple wire is the 5 volt standby. The gray wire is really important. It sends a control signal called something like "PowerOK" from the PSU to the motherboard. It should go from 0 volts to about 5 volts within a half second of pressing the case power switch. If you do not have this signal, your computer will not boot. The tolerances should be +/- 5%. If not, the PSU is bad.
Unfortunately (yes, there's a "gotcha"), passing all the above does not mean that the PSU is good. It's not being tested under any kind of load. But if the fan doesn't turn on, the PSU is dead.
On to the real troubleshooting ...
(The rest of this assumes that you have a case speaker attached to the motherboard.)
Disconnect everything from the motherboard except the CPU and HSF, the two power cables going to the motherboard,and case power switch. Boot. You should hear a series of long single beeps indicating missing memory. Silence here indicates, in probable order, a bad PSU, motherboard, or CPU.
To eliminate the possiblility of a bad installation where something is shorting and shutting down the PSU, you will need to pull the motherboard out of the case and reassemble the components on an insulated surface. This is called "breadboarding" - from the 1920's homebrew radio days. I always breadboard a new or recycled build. It lets me test components before I go through the trouble of installing them in a case.
If you get the long beeps, add a stick of RAM. Boot. The beep pattern should change to one long and two or three short beeps. Silence indicates that the RAM is shorting out the PSU (very rare). Long single beeps indicates that the BIOS does not recognize the presence of the RAM.
If you get the one long and two or three short beeps, test the rest of the RAM. If good, install the video card and any needed power cables and plug in the monitor. If the video card is good, the system should successfully POST (one short beep, usually) and you will see the boot screen and messages.
Note - an inadequate PSU will cause a failure here or any step later.
Note - you do not need drives or a keyboard to successfully POST (generally a single short beep).
If you successfully POST, start plugging in the rest of the components, one at a time.