Short circuits

Anywhere in your laptop, physical or electrical damage can cause a short circuit. A short circuit will consume all available power, causing your laptop to not turn on. If your power adapter LED comes on when you plug it into the wall, but then goes off when you plug in the laptop, you probably have a short circuit.
Troubleshooting short circuits

An ohm-meter / multimeter will read less than ~3 ohms when you measure the resistance between sides of power going into a short circuit. When measureing between then power pins going into the back of your computer, consider the 2 pins where you expect voltage to be applied by the AC adapter. If the resistance indicates a short circuit, this is bad.

A common trick to help isolate possible problems is to try to removing anything that might be a short circuit. Take out the battery, the hard drive, the DVD/CD drive, the floppy drive, PCMCIA cards, USB devices, miniPCI cards. See if the computer will then turn on.

You might have luck with a thermal imager (see story below) to find a hot spot caused by the short circuit.
Spilled liquid

Laptops exposed to small amounts of spilled liquid are repairable in some cases. Liquid can cause the following types of damage:

* Short circuit due to wet liquid (problem goes away after liquid evaporates)
* Short circuit due to dried liquid residue
* Corrosion (especially bad with soda such as Coke or Pepsi)
* Electronic components damanged by above short circuits

Immediately when a spill occurs, you typically want to turn off the laptop, turn the laptop upside down (so the liquid goes out the same way it came in), remove the power and battery, and allow to dry at least overnight.

To remove dried liquid residue: Try removing the keyboard and cleaning any components that have dried liquid, by using distilled water and Q-tips.
Computer turns off randomly