First of all, most of the hardware failures that are going to be presented in this article require you to disassemble the laptop and work on its internal components. Because of this, you need the required components to take the laptop apart (such as screwdrivers, etc.) and space, lots of space. If you don't have a workbench, then working on that large table in the living room should do it.
Take extra careful about ESD (electrostatic discharge). Sure, this is important when working with a desktop system too, but you really shouldn't consider this as some sort of overreaction or waste of time. A few anti-static conductive bags should suffice. These are going to be useful when you take apart components from the laptop and you need a safe place to put them down.
Now let's begin. If the laptop does not power on using the batteries, then try to power it using the AC adapter. Pay attention and see whether the LED on the laptop lights up, signaling that the adapter was detected. If yes, then your power jack works. If the notebook still does not power on, disassemble the laptop and remove the batteries. Look first for the CMOS battery and take it out.
If your particular laptop model has more than one battery, like a multi-port battery or a handle battery, then take those out too. This should serve as a master reset. Try powering on using the AC adapter. If it still does not power on, then read on.
Your best bet is that the problem is memory-related. What you need to do is re-seat the memory modules. If there are more than one, then try swapping them between each other and try different slots. Try booting again. If it does not work, then try running with only one module - this way, you can tell whether one is faulty.
Narrow down the problem by removing each "unnecessary" component. Disassemble the laptop and remove the HDD, CD/DVD drives, and the modem and/or wireless card(s). If possible, take out the sound card module (if separate). To minimize the possible causes, you may disconnect the LCD panel as well. Try running this way. If it does boot, then try connecting each connector one at a time, booting up in between to localize the problem.
On the other hand, if you have stripped it down to a bare-bone laptop and it still refuses to POST, then chances are your motherboard might be the root of the problem. Another option is temporarily replacing the memory modules with good ones. This way, you can eliminate that from the list of possible causes. This leaves the motherboard and CPU. Take apart everything; re-seat both the CPU and motherboard. Try again.
As a last resort, if you are familiar with using a multimeter, then try checking the power button and see whether it actually "switches" or not. You may also monitor the voltages during the VRM circuitry (whether or not it gets power). Verify the resistances between the common places (such as verifying whether the motherboard is grounded or not - you know, those metallic rings near the screw holes).
