To exude tremendous effort, to try extremely hard.

Originally referred to the governor on old steam engines that had two ball that rotated across one another that would control speed depending on how fast they spun. Balls out meant going full speed or power.

Old timey terminology: on steam engines, they had steel balls in there that resricted steam flow. When the conductor was ready to pick up speed, he'd yell "Balls out!", signaling the help to do just that, thereby letting them haul some ass.

This refers to the governor on a steam engine. Two heavy balls are attached to the engine so that as engine speed increases, the centrifigal force of the flywheel causes the balls to rise. As the balls top out, they govern (limit) the engine, thereby controlling maximum engine speed. "Balls out," then, refers to running the engine at maximum speed.

giving your all in a certain situation, being courageous

When things get really tough or crazy, and one has to try harder to accomplish a goal.

When playing beer pong and you miss the table with your shot. As punishment for this embarassing lack of ability the shooter must expose his balls until the next player takes his shot.

This is the real way to play, as opposed to the 'partner forgoes their shot' rule