Can the quarterback run with the ball?

Not directly. The quarterback (the player who takes the snap) can't run the ball across the line of scrimmage.

What the offense can do:

• Hand off, pitch or lateral the ball behind the line of scrimmage, and the receiving player can run or throw.
• The quarterback can then become a receiver if the ball was handed off, pitched or lateraled behind the line.
• Throw one forward pass per play, from behind the line, and it must cross the line of scrimmage.

The offense can't hand the ball first to the center (no "center sneak"). Once a handoff, pitch or lateral happens, the 7-second pass clock no longer applies and every defender may rush.