Meaning: To push to the limit, go all out, full speed.
Example: If you study balls to the wall from now on, you just might pass your algebra class.
Origin: A very colorful phrase, one needs to be careful when using "balls to the wall". Although its real origin is very benign, most people assume it is a reference to testicles.
In fact it is from fighter planes. The "balls" are knobs atop the plane's throttle control. Pushing the throttle all the way forward, to the wall of the cockpit, is to apply full throttle.

Alternatively,

Early railroad locomotives were powered by steam engines. Those engines typically had a mechanical governor. These governors consisted of two weighted steel balls mounted at the ends of two arms, jointed and attached to the end of a vertical shaft that was connected to the interior of the engine. The entire assembly is encased in a housing.

The shafts and the weighted balls rotate at a rate driven by the engine speed. As engine speed increases, the assembly rotates at a faster speed and centrifugal force causes the weighted balls to hinge upward on the arms.

At maximum engine speed - controlled by these governors - centrifugal force causes the two weighted balls to rotate with their connecting shafts parallel to the ground and thereby nearly touching the sides - the walls - of their metal housing.

So, an engineer driving his steam locomotive at full throttle was going "balls to the wall". The expression came to be used commonly to describe something going full speed.

Thanks to Virgil Jose

This may also be an example of rhyming slang.