Wow Bob it could be quite a few things.
I don't know how much you know about computer hardware, so my post below may be too remedial or too advanced, so don't be shy about saying "I have no idea what you're talking about" or "Duh," as is appropriate.
First of all, it is not likely the power supply since the fans are spinning up, so I would not test this yet (though a power supply on its last leg can do wonky things, like spin a fan but not boot a system). Still, statistically I would eliminate this possibility for now.
It is also not likely an issue with the case cables being connected to the mother board (mobo) incorrectly, since whether or not the reset switch, etc is connected correctly does not normally affect the boot or POST.
Be sure the power supply is connected to the mobo with both connections: usually a 20 or 24 pin connector and a 4 pin connector near the heat sink.
It could be the harddrive (I know it worked in another computer, but if I had a dollar for every time a piece of hardware worked in one machine and then failed when I reinstalled it in another...), ram, motherboard, or processor. It could also be a bad cable or ribbon.
Whenever I have a computer fail and don't know why, I generally get it down to the fewest components possible: power supply, mobo, cpu, heat sink, ram, and harddrive (hdd). So, I pull out all video cards, modems, nics, etc and only use one stick of ram, then I try to boot again. If it does not boot, I switch the one stick of ram with a different stick and try again. If it still does not boot, this tells me the ram is probably not the problem.
Next I usually try to reboot with a different harddrive. Do you have a different harddrive to try? If it still doesn't boot, then I try a different sata cable (or IDE ribbon if you are using an IDE harddrive). If it still doesn't boot, by now I am getting nervous it is the mobo. So, I then check to be sure the processor is seated correctly under the heatsink, and I try longshot fixes like a different power supply. If it still doesn't boot, and the ram, hdd, power supply, and cables are all okay, it is probably either the mobo or the processor (usually the mobo).
At this point you have to try the components on a different mobo, which is a tremendous pain since you have to pull everything out. If you can tell me what you have tried so far, I will be happy to guide you as I can, though probably nothing but tedious trial and error will find the source of your problem.