There is no single way to test the motherboard, they have hundreds of components to test. Easiest thing to check is for bad capacitors, happens a lot. They'd be puffy or actually burst at the tops.
Aside from that, you'd have to basically learn how the motherboards work, what part should have what voltage, resistance, etc..., and start checking them. Plus they are usually built in layers, have different other chips on them aside from the CPU. Good project to play with if you really want to do this, but don't except someone to say "just hook up wire A to part 1, wire B to part 2, if you get no reading it's bad".
Failures could be from he power supply also, or the video card. Those two often cause computer issues.