At this point it is either your power supply that is failing or your motherboard. The reason why we can disregard the other components at this point is that if any part fails, your motherboard will give you an error message one way or another by either beeping or giving an error code on the screen.
Most likely your Power Supply is damaged. Don't worry, it is one of the most failure-prone items in PC systems. It is also very easy to replace and does not cost much!
On the bright side, your Motherboard, CPU and RAM should be OK. (It is very rare that the Motherboard gets damaged, specially if you have a good quality motherboard as they have some level of protection).
You need to get a working power supply and try it on your PC. Power Supplies are not servicable.
But if changing the power supply does not repair the problem then you need a new Motherboard.
You can also visit our computer hardware and equipment vendors list, where you can browse for all the major computer parts equipments on the web.
Look for Quality to avoid future troubles.
Do not buy the cheapest ones as they are unstable and tend to burn. Also get at least 450W of power output to have ample power.
This should fix your problem.
Even if you find your power supply working and your computer is still dead (by trying your power supply on another computer), there is a high probability that your power supply becomes unstable under high loads and that is what has burnt your motherboard in the first place.
At this point if your motherboard is fried and that you have to buy a new one do not risk burning it to save 30$ on the power supply (you also need to change the RAM & CPU or risk burning your new Motherboard).
*Warning: You might burn your new parts if you mix them with the burnt ones!