There's pretty much two ways to do the same thing. If you just want the power supply to turn on (like to test hdd spinup or something) then you can check around the net for ATX power supply specification. There's two pins in the plug from the psu to the m/b that, when shorted, turn on the psu.
Alternatively, those two wires, when plugged into the motherboard, are generally routed to another place on the board and stick out as pins. There's an area on the board where you plug in things like the hard drive and power LED's, PC speaker, and case power and reset switches. You'll have to check your motherboard's manual to figure out which ones do what. You should be able to just short the pins where the case power switch is supposed to connect (like with a screw driver), and it does the same thing pressing a plugged in switch would do.
I warn you, you should be sure you have the right pins, else you could screw up something worse than it is.