Broken Computer

As this was listed under Desktop I'm assuming it is a desktop computer so all that should be required is if you know what you are doing is remove a cover and have a look see and use your nose to see if you can smell anything nasty.

Power Supplies are the most common problem in Desktop Computers and can range from nothing more complicated than changing the Power Supply to needing an entirely new computer depending on the quality of the Power Supply.

The better ones sacarifice themselves to protect the internals of the computer where as the cheaper ones pass any problems in to the internal components and kill things like the M'Board.

Most Desktops can be sorted by any computer repairer and they can fault find what has happened here which is probably the best way to proceed as the internals of computers are delicate and very sensitive to Static which the human hand is a perfect conductor of so if you open the case stick your hand inside and do not ground yourself properly you can do more damage without knowing anything has happened.

Your description points to a Capacitor failing which if it's in the Power Supply is an easy and realitively cheap fix but if it's on the M'Board then at the very least it's a new M'Board required.

It's best to let someone who knows what they are doing here fault find the problem particuarly if there is any important data on the system but if you want to play and don't care if you do more damage than you already have by all means have a look and see what you can but you have to be really careful in how you proceed.

The first thing to do would be to open the computer and sniff around the m/b and power supply, searching for the smell of burned electronics. Hopefully the problem IS just the power supply and that replacing it works without replacing the m/b and CPU, graphics card, or RAM. I suspect the easiest way to diagnose (beyond your nose) would be to get a new power supply and install that and then peer on just the barest amount of things, before trying to replace the m/b, CPU (and cooling) GPU,RAMand maybe the HDD, and see if that gets response. If not then something more serious was damaged (most likely m/b or GPU. If it starts working, add in any other components 1 at a time.
For the rest of the system, lacking special diagnostic told, if you have another computer, swap in the other components to the hot machine one at a time to you see what doesn't work until you identify the culprit. This is easier) and cheaper) than buying new components one at a time till it starts working