Thanks for specifying. Not sure why I assumed a laptop, I guess I was guessing at the model #.
So, this is what you're getting so far:
Monitor seems to recognize the computer turned on, but nothing else does.
Very first, an important note: I'd highly suggest taking pictures of where everything is and/or documenting it as best you can. That way, when you're putting it back together, you can make sure that you put everything back in the original place... if you don't end up needing to change one of those. The main point is that you'll have a visual point of reference.
Now, this'd be worth the learning process, anyhow: I'd go ahead and disconnect everything except for the processor/heatsink/fan, and the power connector(s) from the power supply.
Here's a review I found on your desktop. You're looking at technology from 7 years ago, at least, per the review:
My initial guess for this is that the power supply and/or motherboard is dead or dying. But that doesn't mean it's definitely the case.
If you can remove everything from the case except for these:
Power Connections (connections from power supply to motherboard)
Processor, heatsink, and fan - don't move those just yet.
Front panel connections - this is where your power button, reset button, hdd led(s), speaker, etc typically connect. You want to leave these in place if they are all connected.
Also, unplug the power connections to your drives - hard drive, optical (cd/dvd) drive, floppy drive, zip drive if there, etc...
This also means pulling out any add-in cards - PCI, AGP, etc. Anything plugged into a brown/black/white slot at the bottom of the case most likely... or for that matter - anything plugged into any slot at the bottom of the board.
Even go ahead and unplug the power connections from any case fans. Only leave the power connection from the motherboard to the fan on your processor/heatsink.
You don't have to actually take your drives OUT of the case, just disconnect them for now.
Once you have everything disconnected (including RAM), as metnioned above, keep your fingers clear of the motherboard, fans, etc, and try to power up the machine while watching the fans, etc. Also listen for beeps. If you're motherboard is getting enough power, and you do have a case speaker, then it should at least beep telling you the memory is missing or unrecognized. The fan on the CPU should be spinning.
If you don't hear beeps, then either you don't have a speaker, or else it's definitely the power supply or motherboard.
Do you happen to have another desktop computer of any sort to borrow parts from for testing? If so, you can have a much better chance of finding the issue.
Oh, also, you can visually inspect the motherboard, to see if you have any bulging/busted/leaking capacitors:
If you do have those, then unless you've got the right equipment for soldering this, and can get some replacement capacitors cheap, then you need a new motherboard. You can probably find a used one cheap on eBay, I'd imagine, that's identical to yours. If not, you can probably find a P4 motherboard on eBay or elsewhere that will work for cheap. It is old hardware, so i'd not spend more than $30 on a used mobo, total.
Anyway, glance over these poorly placed thoughts, look over your system as best you can, and post back.
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"If to err is human, then I must be some kind of human!" -Me