Do you have an official Arduino Mega 2560 board ? http://arduino.cc/en/Main/arduinoBoardMega2560
It should turn on when power is applied to the DC barrel jack or to Vin. If it doesn't, perhaps the connector doesn't fit, or the power supply is broken or the voltage regulator on the Arduino board is blown. The DC barrel jack input is protected against negative voltages.
Can you try another power supply ?
Do you have a multimeter ?
I checked and the power is 12 volts and I confirmed that it is 12 volts on the terminals on the board. So I know the power is tranfering to the board. Not sure what is happening after that.
This is a brand new board and it was first connected by USB, then I tried direct. I made sure that the positive was correct before it was connected with a multimeter.
Is the Vin also 12V ? If so, the voltage regulator is broken. Can you see something on the board, like a brown/black burned copper trace or a component that got too hot, or bad soldering ?
The path is like this: Power supply to barrel jack -> diode -> Vin -> voltage regulator -> 5V.
When Vin pin is 12V, and the 5V pin has nothing, there is only the voltage regulator in between, that is the broken part. Or the voltage regulator is not properly soldered, for example when you bought a cheap clone.
Thank you to you guys for your help. Based on what you are telling me I think the regulator is shot. I will get the supplier to send me a new board and run it off the USB for now.
If the AMS1117 is blown (the voltage regulator next to the DC barrel jack), and the rest is okay, you could remove the AMS1117 and still use the board with 5V.