My Arduino works fine when its connected to the computer via the USB but as soon as I disconnect it from the computer, it goes off. I am using a 12V DC adapter (500mA) (center positive) as the external power source. Not sure what is wrong. Please help!
Well with just the 12V DC adapter plugged into the arduino (no USB connected) take a measurement with your digital multimeter between Vin and ground pins, then take a measurement from the 5V and ground pins. That should tell you the story, if your adapter is bad or the on-board +5vdc voltage regulator is bad.
If you don't have a DMM then your first step is to go buy one, there are very inexpensive ones available and one should not be playing with a arduino without having and knowing how to use one.