The most common issue, is the power jack and/or the power adapter. Make sure the jack (socket) is not loose inside the computer; they often have broken solder, or are broken from someone tripping over the cord. Also, check that the plug is the correct one for the laptop, and fits snuggly. Sometimes the wires can be broken inside the insulation in such a way they the connect intermittently. Again, this is usually caused by someone tripping over the cord. Sometimes the wires are damaged by twisting them too much.
If your laptop has an indicator on the front that lights when it is off and plugged in, try gently wiggling the plug in the socket and see if the light blinks. You could also bend the cord to see if the is an internal break. Also, be sure the high voltage cable is plugged tightly into the power adapter "brick."
If you can get the power light to come on, and stay on, try charging your battery with the power off over night. In the morning, make sure the light is still on (though it will have likely turned from amber to green). Start the laptop, and verify that the battery is at 100%. Disconnect the power cord, an allow it to discharge until it reaches less than 20%. Charge and discharge it 10-20 times. This is called exercising a battery. Keeping a laptop plugged in causes some batteries to develop a memory:
http://en.wikipedia.org/wiki/Memory_effect
And low temps can affect batteries and electronics, so allow a cold computer to acclimate before plugging it in or using it.
Good Luck