How do I build a open computer without a case?

Each to his own but why not buy a pc case? You can get them for very little money. Anyway, in answer to your question you can power up the pc by shorting the two pins which normally have the socketed end leading from the power button plugged into them. They will be in slightly different places on different motherboards but are usually marked up clearly enough. The motherboard manual should also have a diagram showing the pin layout.
 
It's not a good idea to build a computer without a case. Dust gets in and harms your computers. The reason he's not using a case is because he is running a test bench which means he has to switch out parts frequently and quickly. Some high end motherboards have a power buttons on the board itself but you can touch two pins together with a metal object to manually turn a computer on.
 

No, dust can't harm your computers. Accidental spills on the other hand...
There's also a risk of hitting something and breaking a piece of hardware off, sneezing and accidentally spitting all over it and so on :)

Either way, you can get a case for $20 as getochkn said.
 




Dust degrades your computer. Problems overheating and part failure.
http://www.wikihow.com/Clean-the-Inside-of-a-Computer
 
in the video that guy is actually using a computer (a open computer case)
some are more expensive than "closed" computer cases
you can always just lay the motherboard on a table and hook up everything to it and just place it around the board :)
 
That'd be 'Cool' for al of about 5 seconds, and you will get tons of problems. It'd be to easy for a pet, or some food/drink to spill and destroy it all. My first PC's power button broke, so i was using a screw driver to short the 2 circuits together for quite some time. Some MoBo's Bios's support Power on by Keyboard, or mouse, snoop your bios for that, u might be able to do that.