Can a Motherboard be Swapped out with a different type?

I’ve never built a computer from scratch so I’m reverse engineering here and need to know what is and isn’t workable.

[b]The Problem:[/b]
The video adapter does not work. It is attached to the motherboard. A PCI video adapter also does not work with this motherboard. I think there is a shorted circuit somewhere on the motherboard. (The hard drive is fine and has been reset to factory settings. All other hardware are also a-ok.)

[b]The Question:[/b]
Can I replace this motherboard (a FoxConn) with a similar type motherboard but most likely a different brand? Because this computer is an HP I am thinking I will not find an exact replacement so will have to opt for the cheapest board that has similar ports.