The input voltage required by the board is 5V only. Since the USB cable supply 5V that isn't a problem because the voltage supplied meets with the Galileo requirements. The same thing happens when using the DC barrel jack to power the board. You must use only a 5V power supply. The problem is in the current drawn by the board. The specs recommended for the DC power supply are 5V output voltage and 3A output current. That is because the board can be demand up to 3A, that will depend on the application and the external devices attached to the board.
The USB ports on a PC can only supply 500mA. If you power the board through the USB port and the board demands more current than the USB source can supply, the board can be damaged. In the best scenario the firmware can be corrupted only, but in the worst scenario the board can be damaged permanently. Same happens if you power the board through the DC barrel jack with a power supply which has an output current lower than the recommended.
My recommendation is to use always the DC power adapter to power the board with the recommended specs; voltage and current. If you don't have one, try to get a new one because you could damage the board if you keep plugging it through the USB port.