Well, the thing goes like this: I had my PC on, and it was running, it was left to run over-night, and suddenly, yesterday, when I turned on the monitor, all, that I saw, was a mish-mash of colors. So I restarted the PC.
Now, after restarting it, it constantly beeped in this way: a long beep, followed by two shorter beeps. Since my PC's BIOS is an Award/Phoneix BIOS, this combination of beeps is supposed to mean, that there's a graphics card problem. Having an on-board graphics card, it definitely can't be a slot problem anyway.
Now, it did that for the whole day. I thought, that the on-board graphics card permanently broke, so today, I bought a new graphics card. But then, once I arrived home, I turned the PC on, just to test, and guess what? It was suddenly working properly again, with the original on-board graphics card.

Now, I'd have a few questions:
1. What are the odds of yesterday's problems repeating?
2. Does this mean, that the on-board graphics card is defective?
3. I remember having had a lot of BSOD's in both Windows 2000 Professional, and Windows XP Professional, when I got this PC, until I slowed down the memory speed in the beginning, and the BSOD's were all related to the nVidia graphics driver (note: the on-board graphics card is a nVidia GeForce 6150). Is this in any way related, to what happened yesterday?
4. Is it better, if I return the new graphics card back to the shop, where I bought it, or should I rather keep it?

Thanks in advance.

Well this is a problem I sometimes have with my nvidea graphics card. im not sure what causes it but i think that it may have something to do with overheating or the buffer. Now, it doesnt give me fancy colours but it does freeze at the current screen. It happens rarely but to solve it you need to turn the computer off and leave it off at the switch for 10 - 20 mins. That usually works.

And keep the graphics card. Inbuilt graphics are not renowned for their performance.

Well, that pattern of beeps is a video related fault. I have known onboard video cards to go faulty (and in some cases like yours - intermittently faulty). Onboard video cards are, after all, still usually hung off the PCI bus (or AGP bus) that everything else would normally attach to. Corrupt displays with flashy colours I've usually found to be memory related (I used to repair these types of faults by replacing the surface mounted memory!). If your particular board uses shared video memory (ie. it eats away at your main memory) then I'd investigate this first.

If not, then try disabling the onboard video and run with your replacement card for a while. Doesn't hurt to try!

Download BartPE or UltimateBootCD and run a memory test.

The problem was already discovered to have been over-heating, due to big amounts of dust both on, and inside the PC.
I cleaned the PC (both outside, and inside), and had no such problems since then.