When I press the power button on the computer it comes but the screen just stays black.
Do you hear confirmation the computer has booted (usually a BIOS single "success" beep)?
Does a power indicator lamp show on the monitor?
If the monitor has an independent mains connection (mains is supplied from a wall outlet, not a socket on the computer), will the monitor power up from its own power button?
If the computer beeps (normal boot up) and continues to load Windows (hard drive activity and startup sound), but there is no power indication on the monitor, or the monitor does not start up and show a splash screen/no input indication when you press its power button, the monitor itself is faulty, and as Chris has suggested, you can diagnose things like blown fuse.
If the computer does not confirm boot-up, and the monitor shows a power light but the screen remains black, this suggests the computer has developed a hardware fault.
If the computer beeps, continues to load Windows, there is a power indication on the monitor but the screen remains black, it may be the backlighting on the monitor has failed. If this happens, you can normally make out a faint screen image if you darken the room and hold a torch against the screen.