Well, yesterday I was minding my own bussines, browsing and watching movie and suddenly my PC just turned off without any warning or indication. At first I thought it's just power outage but monitor and speakers were still on. When I tried to turn it on again it "tried" to start but immediately ceased. Fan moves slighty and stops again.
Don't you know what could it be? Motherboard or power source? And is there any hope to repair it?
Also I was messing with cables and I pulled out cables that goes from start and restart buttons to motherboard. One of them is splitted into six parts and I have no idea, how to put them back. They are of different colors as is plug but I can't figure out how to connect them. Mainly because I have no way to test if it works because it won't start anyway. But at first the fan tried to move and now it does absolutely nothing. But it can be either by wrong connection or just cause it's broken.

The most likely candidate is the PSU. I'd try replacing that with a branded model. Ideally of somewhat greater wattage than the current one. If it's a pre-built that would protect you against manufacturers skimping on the PSU, putting too great a load on it. It would also allow you more scope to upgrade in future.
Corsair offer pretty good value for money PSUs with a decent warranty.
As for the power button cables, consult your motherboard manual. Download one online if you don't have it. It's varies too greatly from mobo to mobo to explain it here.

Unless you know what you're doing, you shouldn't be yanking out cables and prodding motherboards. Take it to a repair shop.

What's your PC spec?
I also suspect it's the PSU, because as you said, the fan stops moving. If you're planning to buy a new PSU, I recommend Silverstone, they're rock solid.

/\ This.

Well, I know I shouldn't do it but at first I was hoping it is something as simple as loose cable so I tried this. I have at least minimal knowledge of handling of some other parts of PC but I never messed with mobo neither I intend to anymore.
Anyways thanks for advice. I was afraid it is PSU but I always expected that these die somehow more spectacularly. :)
Now I am not sure whether I should buy new PSU, because it's old PC and I was going to buy new one. But it was supposed to happen in few months, when more money comes by.
I can't give you exact specs. I don't know them by heart, I can't look at my PC and documentation I have at home. But it's AMD Athlon 64 3300+, with Nvidia GeForce 6600 GT, 2x 512 RAM and Seagate 160 GB HDD. It's all I can give off the top of my head. :(

I vote for a new PC . . . =)

If it's an old crappy one then this is the perfect excuse to upgrade.

My thoughts exactly . . . =)

Yep. But then it could be a relatively new one...

That happened after I replaced my PSU, because the previous one quite literally exploded. I first replaced only that, but it didn't seem to help. Then, I replaced the mobo, CPU and RAM, which worked. Dunno if anything, like the RAM survived, though.

Time for a new rig. *nods*

If a dud computer doesn't explode, it just isn't right.
Mind investing a bit of... of... Czechs... cheques... money for some C4?

I get this a few times after an outage with my PC. I know it wasn't an outage, but it's worth a go.
Sometimes if the PC turns off without warning, it won't get to POST. It will whir for a bit and then just stop.
What fixes it? Turning the power switch off at the back and turning the mains plug off and back on again, turning the switch back on and then powering it up.
Try that.
Power switch off
Mains off
Mains on
Power switch on
Engage