Something is shorted out.
When you connect via USB, the USB port cannot supply enough current because of the short. The output drops to 3.9V and the USB port on your computer reports an error, probably due to over current.
When you connect the external power supply, it has enough power to supply the short. The output remains at 5V.
You can think of power supplies as having a very small internal resistance (R_p), and the board as being a resistive load (R_b). This forms a voltage divider, but because R_b >> R_p the voltage seen by the board is basically the same as that of the power supply. With a short, R_b is often < R_p, and thus the voltage seen by the board is much lower than the power supply voltage.
The fix will depend on which component is getting hot. Plug in the external power supply and check which one it is. Note: The component will get hot enough to burn your finger after a few seconds. One way is to check one component at a time, disconnecting the power between each check. Start with the big ICs.
If either the Atmega328P or the smaller ATmega8u2 (?) are hot, they will have to be replaced.
Finally, boards don't short out by themselves. Something is wrong with your circuit or you may have placed the board down on some conductive surface. If you post you circuit we might be able to spot the problem.