First allow me to explain, I was simply in the process of cleaning out my old PC. I was using a can of compressed air to blow out the accumulation of dust from within. Once done, I made sure all plugs were snuggly in place, and closed up my PC. I turned it on, and it acted as it should, however, there was nothing displayed on my monitor. (It acted asthough it wasn't connected to the PC at all).

My PC is a bit old. Here are the specs I could find.
Its a 2002 Sony Vaio PCV-RZ22G. It runs windows XP
Intel Pentium 4 Processor
80 GB 7200 rpm hard drive
512MB DDR333 (PC2700) SDRAM
NVIDIA GeForce4 MX440 graphics chip w/64 MB VRAM.

Some notes. It didn't beep upon startup, but it usually would. The power lights all turn on, the HDD lights turned on. Both the fans turned on, but it displayed NO image on the monitor at all.

Any and all help would be greatly appreciated, and thanks in advance!

Hi and welcome to Daniweb,
Normally the problem is caused by faulty seated RAM, note I say normally, there are many possible other causes. Try by removing your RAM, cleaning the contacts with a pencil eraser and putting in another slot. Ensure that you earth yourself to the metal box and turn off at the power before you touch the RAM.
Your computer is old but it should work well if you don't want to play high usage games. If you are not going to buy a new one in the near future, I would recommend you increase the RAM to 1GB if you can afford but again I say, it is OK as it is.
If you are still having problems after cleaning RAM, go back and unplug and replug everything again.
If you need more assistance, please post back.

After doing everything you suggested, the situation remains the same. I also remembered that part of the processor fan (the mechanism that holds the fan and heat sink in place) is broken, there are 2 parts that hold it down with little hooks that go over top of the heat sink, 1 of the spots the where a hook would fit is broken, so it can't be held in place by 2 of the hook devices, only one. Not sure if this could cause the problem or not... But to my knowledge its been broken for a while, i only worry about the processor falling out, lol...

Additional problems I have had with this PC recently are it randomly shuts off, and gets very hot. I assume this is because of the afforementioned heat sink and fan problem..

So any further suggestions?

As I said in my first post there are many other possibilities, I would be looking at a repair tech to give you a free or low cost quote.
In regard to your other problem, you are correct Heat would cause that problem and it is most probably caused by the lack of connection between your heat sink and CPU.
As your motherboard is old, you may have problems finding a replacement part. You may have to look at a new upgraded motherboard, which also means a CPU, memory and probably, video card (if you want), PSU. While you are at it you may as well get a new hard drive. Let me see... that is a new rebuilt computer.

I think the cpu would not overheat excessivelyif there was at least some contact with the heatsink... you should at least see the BIOS run before any overheating occurs, but loading windows is a little cpu intensive. Your monitor.. when you power it up but not the pc, does it self test... ie, display its name and then perhaps say something like No Signal?
The designer of Intel heatsink hold-downs should be taken out the back and dealt with severely...
You didn't knock your graphics card loose?

lol, I'll probably not even bother if I have to rebuild a whole new computer XD

When I turn the monitor on, it says: "Information: No Input Signal, Go To Power Save".
Also, I made sure to carefully remove my graphics card, make sure it was clean, then put it back snuggly in its home. I also made sure I was well grounded to the computers metal box before touching any vital component.

I assume that since that model comes with a graphics card as standard that there is no mb integrated graphics, and thus no graphics port on the mb itself to try?
Okay... perhaps BIOS is not starting. Try resetting it to defaults - to do this remove the CMOS battery and replace it after about 5 mins; your guide may tell you of a shorting link to place temporarily in another socket instead. The CMOS battery is about 20mm in diameter. Please turn off and disconnect power to the machine first.
Check again power supply connection to the mainboard.

I unplugged the CMOS battery for about 10 minutes and plugged it back in. I also disconnected the power supply from the motherboard and reconnected it. The situation remains unchanged. It doesn't seem to be running BIOS though. I waited a bit and tried to enter my password at the user login trying to "go blind" to see if it would make any sounds, and I got nothing. So the system turns on, the CD and DVD drives eject. All lights flash as normal. The PC acts like it should, but its apparently not running BIOS, it doesn't have the customary "beep" upon being turned on, and has no display to the monitor.

Within 5 or so secs of power On BIOS should be addressing the hard disk.... the hdd lamp should flash intermittently, you should hear the head motor twitching. If not, it is a mainboard or RAM problem. I see you checked RAM seating; it may be time to borrow a stick from your local repair shop or dealer. The local nerd will have a selection. Keep to PC2700 SDRAM.
If yes, then I suspect the graphics card.
You might try unplugging the hdd and trying to boot - it will not get far, of course, but if it was the hdd pulling the sys down you would at least see/hear some errors. But my money would be on the RAM. It does die, and when it does nothing happens.

well, the HDD lamp flashes, and the motor gets goin. the fans start to run, all that good stuff... i'm starting to think it may be the RAM going out... hmmm...

When you have compressed air, you end up with condensation. There is a chance that some moisture has gotten in where it shouldn't have and made something go pop.