First, make sure the power plug is fully into the computer. Stupid, but check it. Also a lot of PSU's have an on off switch, make sure it is switched to on.
After you've checked those two things; we need to determine what the problem is. I had a similar problem, and it ended up being a fried motherboard, (no fans or lights even spun up). This is rather rare, so we'll check the possibility of a fried motherboard last. B/c usually you will at least get some fans to spin up even with a fried motherboard. To isolate the problem to a specific component of the computer I need you to do a few things:
First, to ensure that it is not the PSU, I am going to need you to short the PSU. This is very easy, and a very effective way to test your PSU.
To do this process follow the steps below:
1. Straighten out a narrow paper clip, (a bobby pin, or staple will also work if you can straighten it out).
2. Detach the molex cable running from your PSU to your Mother Board.
3. Place one end of the straightened paper clip into the whole on the molex cable with the green wire coming out of it. Attach the other end of the paper clip to a whole in the molex cable with a black wire coming out of it. There are several black wires, and it does not matter which one you choose. The idea here is that you complete the circuit between the green wire and black wire by using a piece of metal to make contact with the metal contacts in each of the wholes.
4. If your PSU spins up to life, then the problem is not your PSU.
5. If it does not spin up, and your sure your making contact with the metal contacts in the wholes, it is your PSU.
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Now in the event it is not your PSU, there are some other reasons as to why your computer will not turn on even though you hit the power button and your PSU is good.
One of the simplest to check is that your power button may physically be broken. If you have played around with what metal pins on the mobo the 'on-switch' connects to, now would be a good time to tell us so. B/c if you have you probably reattached it to the wrong set of metal pins.
If you haven't, then please check to make sure that the cable going from your onswitch to the pins on the motherboard is firmly and securely attached. And if it is, try reseating it. Take careful note of EXACTLY which metal pins it is connected to before reseating it. Also check to make sure that on the other side of the 'on switch', no cables have become loose or frayed. When I say other side of the 'on switch' i am referring to the part of the 'onswitch' that is physically pushed in when you push the power button.
If you determine that the problem is not the power button, it is possible that a component in your computer is preventing a sucessful POST. While unlikely, in my mind this is the next most likely problem. To test this, remove the RAM, Video Card, harddrives, and cd drives. Only if you feel comfortable doing this. I am not sure how comfortable you are with computers. Try and boot the computer now. If it does not boot up still, then it is not any of these components that are the problem. If it attempts to POST, and your computer makes a beeping noise; then one by one reattach each component and attempt to start up the computer in between attaching components. When it stops booting up, this is your broken device.
If all of the above fails, try resetting your CMOS. Consult your motherboard manual, (or if you purchased this computer from DELL or GATEWAY, ask them how to do it). There is usually a jumper near the CMOS battery to activate, (your manual or tech support will explain how to do this, as it varies by make). This will often fix many problems.
If that also fails; then I am afraid that the next logical conclusion is that there was a surge in electricity, (either ESD or power spike), that fried one or more of your components in your computer. You would have to individually test each component in a different machine to see what still works.
