That's just it. The only thing I don't have is an extra PCI video card laying around. I have a bunch of extra parts, even power supply units, but no extra PCI video card.
When I unplugged everything except the power, fan, and video card, I got a NO SIGNAL on the monitor....must be video card then, huh?

you still had cpu and memory installed too, right? If you didn't, it doesn't tell you anything. If you did...it could still be any of those parts. You have a spare cpu or RAM module you can swap in?

Let me throw in a little list for testing.

These things should be connect and plugged in.

video card (if your video card requires power make sure its plugged in)
memory
cpu
power supply (plugged into the motherboard)
montior (plugged into the video card)

That is all you should have connected to test. Everthing else should be disconnected.

If you still get the no signal then your problem lies with the
video card
memory
cpu (unlikly but possible)
power supply (unlikly but possible)

Your best bet right now would be to take it into a repair shop so they can tell you what part is bad. If you have a Best Buy close to you they have a little repair shop in their store.

Other things you can try.
If your video card has a fan on it make sure the fan is still turning. Touch the video card to see how hot it is when the pc is on.

Double check to make sure your cpu fan is turning as well.

Double check to make sure your power supply fan is turning as well.

Unplug your video card and plug it into a different PCI slot, you might have a bad PCI slot.

Thanks, ElevenBravo:

I've done what you suggested above already and have no signal on the monitor. I just don't understand WHY you still wouldn't hear the usual "booting up" sounds from the computer even IF the video card is bad and there's a "no signal" coming from the monitor screen if that's all that's wrong with the PC. What I mean is this:
IF it is the PCI video card and only the video card, wouldn't you still hear the usual grinding, booting up sounds from the PC until you received the "beep" sound, indicative of something not connected correctly?

Yes, that is correct. You still should hear a beeping sound even if your video card was bad. Which makes me believe you have a motherboard or ram problem. Can you tell me what brand of memory and what brand of motherboad you have?

250 powersupply, you really must hate yourself, slightest fluctuation in power and its gone. the absolute bare minimus should be 350W and of quality manufacturer. This is usually the first thing that needs replacement.