How do you find out the details of your laptops motherboard?

I assume you're asking "how do I know what kind of motherboard I have" and not the usually much easier "which part in the computer is the motherboard" (it's almost certainly the biggest circuit board in there).

The only sure one I know of is to look for the manufacturer's name and model number silkscreened onto the board somewhere.

Alternatively, if it's hooked up and working, the power-on self-test may tell you, though sometimes these are a little more vague than might be hoped (for example, while it's likely to tell you the manufacturer and possibly the model, it may or may not specify the particular revision).