Your battery may be on it's last legs...is this happening when it's plugged into the power cord?

Yes.

It could very well be due to overheating. The fans and intakes are probably clogged with dust. You could try to clean it and see if that helps. If you have a compressed air spray, that works very well. Still, you'll have to take it apart to get at the insides of it.

:/ I tried that actually. If overheating were to damage something, what would be the first thing to fry?

Overheating can cause damages to many parts not just one but if anything would go first it would more then likely be the cpu or ram

But try this, take the battery out and just have it plugged in, and see what happens if its a battery issue then the system should run fine, if its a power cord issue the system will not turn on at all.

Somethings to also remember is even though batteries don't have memory effects anymore, having a battery and power cable plugged into a laptop for extended periods of time will put a wear and tear effect on the battery, meaning when the battery is fully charged it will drain itself to protect itself from overcharging; it will continue that cycle over and over again. Protect your batteries and don't have it in your laptop unless you're on the go, go.

If you're a bit tech-savy then you can take it a bit apart and carefully blow the dust out of the fans. It should help with better cooling. It is possible that the fan is not working properly and then it can overheat pretty quickly. Normally modern CPUs have protection against permanent damage so with a bit of luck you won't loose anything.

Bingo. One of them isn't going at all

Another possibility is that the computer has been compromised on the software side (e.g. someone installed malware while you left it unattended). You could try booting a LiveCD to see if the system runs normally (when loading the OS from the CD and not using the HDD) - if it does, that suggests that there is a software (or HDD) problem.

http://www.ubuntu.com/download/desktop/try-ubuntu-before-you-install
has instructions for running Ubuntu as a LiveCD

Or if you're comfy with a command line,
http://www.sysresccd.org/SystemRescueCd_Homepage
is a great liveCD that is nice to have around for diagnostics, data recovery, etc. (also, it's smaller than Ubuntu, so quicker to download).


EDIT:

Also, if an overheat is causing the shutdowns, you may have entries reflecting that in the Windows error log, if you can keep the computer up long enough to check that. Overheating components can generally throw some errors when they get into the danger zone, before the whole thing locks up - and the error message will ID the overheating part (although the cause of the overheating might be a dead fan, or fried thermal paste from a bad heat excursion).

Fans are relatively inexpensive to replace in most machines.

On the other hand older HPs were prone to video card errors after overheating, still fixable but you need to take apart the thing and literally put the motherboard in the oven for a couple of minutes. Like here. Be sure to study a DIY on how to take apart/put together the laptop before and a picture of every step is also helpful.

Cheaper to replace a fan than it is to replace a whole Laptop, no?

I would take that as good news if I were you =)

Ah, yeah that'd do it. I hope it's one you can easily replace. At least you definitely know the cause of the problem now.

That's a start. You should probably Google the disassembly instructions (unofficial ones are surprisingly readily available for many models) to see if it's going to be a big job, because there is no way of telling in advance. I have heard reports of laptops where the fan is one of the easiest components to access, but on the other end of the scale is my old HP, where parts overlapped so much that when I wanted to remove the fan, I had to disassemble absolutely everything to the point that only eight screws (all of them pinning down the heatsink) remained fastened when I was done.

Paying a computer shop to do it is so expensive that it's probably out of the question anyway.