Check the underside of the RAMPS board (removed from the Arduino Mega) for any burnt tracks. This is a good indicator.
Next, there is a PTC fuse (F1) that covers all the motor electronics, and the D9 and D10 outputs (usually Fan and Extruder). If your motors work, then the fuse is fine, otherwise I'd suspect the fuse has blown (they're NOT supposed to be able to blow, but I've seen it happen).
Next, check the FET that runs D10 (labelled Q1). You need to see if when the extruder is switched on, there is a voltage at the gate of the FET. You can do this by measuring the voltage at R13 (it's a 10 Ohm resistor, right near LED1). It's small but you should be able to get a voltmeter probe to it. Compare against ground (eg: -V or COM probe to the -V input to RAMPS from your PSU).
- If there is ~+5V on both sides, then it's the FET (failed gate open circuit).
- If there is ~+5V on one side and zero on the other, you may have blown the resistor (though this is unlikely) but it's more likely the FET (failed gate closed circuit).
- If there is ~+5V on one side and something non-zero on the other, then the FET may still be dead (failed gate with some resistance).
- If there is no ~+5V on either side, then you may have an issue with the Arduino Mega.
To test the Arduino Mega, you'll have to take the RAMPS board off and run a test sketch that sends the pin for D10 high. (Digital pin #10).
If you're not using a Fan, you can always change the firmware to swap D10 and D9 around in the firmware (changing pins.h in Marlin for example). You can then hook up your extruder to D9 and it should just work.
Note: The LED's for Extruder/Fan/Heated Bed all run from the output of the FET. If they don't work, the output is unlikely to work (unless the LED has died). This means you don't need the outputs connected to test everything, which is useful when you're trying to figure out an issue.
PS: Never disconnect/reconnect cables or boards while the power is on - always turn the power off first. This is specifically relevant to motor connections and driver boards, but it makes sense to do it with pretty much every cable into a RAMPS board, just to get into the mind-set of always doing it.