Shame on you two! How's anyone supposed to learn how to fix their own stuff and get out from extortionate and frequently incompetent repair shops if they sit there all scared of breaking something? What's next, buying a sweet gaming machine from HP with a 2 year old video card and 2gb of ram?
Vitek mate, don't listen to them, pull bits apart and look for the problem (I'd also tend to blame the PSU, I thought the CPU might have been the problem until you described the fans trying to spin). Downloading the motherboard manual would be a good move (especially with the cables you yanked out), the motherboard usually has the manufacturer and model number printed somewhere on the board in reasonably large letters

In other news, man dies from an electrical shock in the Czech Republic, tries to fix his home PC. His last words were: "Bzzerghheoeeeroozzz". We have no czech translators here, therefore no translation is available. Škoda. Polls say: Australia to blame.

He was obviously trying to say "Bereghost", wanting to play Baldur's Gate.

xDD thanks for the laughs!

Before you can do anything else, you have to identify the motherboard. Manufacturer and/or Model number should be somewhere on the board itself. Look between expansion slots, on the corners, towards the center, etc. Between writing numbers down and using google, you should come up with it fairly easy.
After you identify the manufacturer and model number of the motherboard, you need to download the pdf manual for your board from the manufacturer's website. Until you have this manual you are shooting in the dark and I would advise against that given your limited knowledge in the area.
The next step would be to reconnect the motherboard's "front panel" connectors according the diagram in the manual. Before you go any further make sure the power is unplugged since you can't verify the integrity of the PSU. Those front panel connectors were probably the wires you disconnected. These contain the power switch, reset button, power LED, HDD access LED, and internal speaker if your case has one. You must verify that those wires are plugged in properly before you can troubleshoot any further considering the power switch is in that bunch. The front panel's connectors should be properly labeled on the wires' connectors or somewhere on the wires themselves. Although it is very important to plug those wires in their proper connector to avoid possible damage, the connector itself can be reversed without any damage as long as it's reversed on it's proper connector on the motherboard. If this happened, reset and power switches would work but the indicator LEDs would be unlit. In that case, you could reverse the indicator LEDs to make them work properly. Again, make sure power is unplugged when connecting those wires.
Next you need to verify that the PSU power is properly connected to the motherboard. There will be a diagram for this in the manual as well. Also check the hard drive data and power connectors to make sure they're secure along with any expansion card you may have (like video card, sound card, etc.)
After this you can actually start troubleshooting the problem. For starters I would try different electrical outlets in your residence. It's possible that you could have faulty power at individual outlets, especially in older wiring.

Hilarious! I'm plussing your post.
And I'm all for poking and prodding, provided you have some clue about how to fix things. If you don't, especially when it concerns the motherboard or, as seems more likely from the majority of posts here, the PSU, I would recommend taking it to a shop. And the people who work there don't need to be incompetent. There are a few good outfits out there, who have staff that are actually knowledgeable and competent. That excludes most big manufacturers' help desks and service centres, obviously...
And as it's an old computer, I would second the notion that this is as good an excuse to upgrade as any, especially if you were planning too, anyway. (And, edit, regardless of whether or not you get this one fixed--it can always serve as a backup.)

Hello guys,
thanks for your advices and "other" input. ;-) I finaly got to another computer and am able to be online for a longer time than quick peak at work. I downloaded manual and managed to connected back all cables. Now I tried PSU from my sister's PC (it's stronger one, 400W compared to my 350 Watts) and it still doesn't work. So it's definitely not the PSU. I tried to disconnect everything else and only things remaining are CPU and mobo. Is there any way to determine which one is it? The mobo is more likely I guess? suppose I am quite screwed here and have to change it if I want to keep this PC?

It's often difficult to distinguish between a CPU and motherboard failure unless you have a spare CPU to swap out or if you can see physical damage (e.g. leaking or bulging capacitors on the motherboard). However, in this case my money would be on the motherboard. Also, unless you got your computer fairly recently a motherboard failure is generally a good time to just do a full upgrade on motherboard, CPU, and RAM.

I had this problem awhile ago and it was actually the RAM. Try just one stick of RAM or get or borrow other sticks. When I got a new set everything worked as normal.

Unfortunately I tried to remove them too and now I also tried to replace them by others and still no luck. I really disconnected everything not needed to switch PC on (only Cooler, PSU and start cable remained connected to motherboard).
Btw. I already know my specs, some are above and remaining that matter are PSU, it's Fortron ATX-350PNF, and mobo, MSI k8N Neo 4 (ver. 1).