Why won't my laptop turn on?

One day I was using my laptop off the battery, when suddenly it shut off. I thought that the battery had just died, but after plugging it in, it didn't work. The charge light goes on and so does the power light. After pressing the power button, I can hear the fan run for approximately 3 seconds, then it shuts off. Nothing shows on the display. I know that the display isn't broken because I plugged it into an external display, which is what I normally use anyway, and nothing showed. It's not the hard drive, because it boots in my other computer. I am very good with this stuff (computer repair is my side hobby), but I have no idea where to start. All I need to know is what is wrong with it so I can replace it.