The sounds you hear are just mechanical waves traveling through the air. The complex apparatus inside your ears translates these waves into signals that travel to your brain, and the round window is part of the machinery of your inner ear. To understand what the round window does, you first have to understand how the inner ear works.
Hearing starts in the middle ear, where incoming sound waves cause your eardrum to vibrate. Several small bones transmit the vibrations from your eardrum to a membrane called the oval window in your inner ear. Behind the oval window lies a fluid-filled, snail-shaped tube called the cochlea. Vibrations in the oval window generate pressure waves in the fluid of the cochlea -- a little like tapping on a sheet of cellophane to create vibrations in water behind it.
The pressure waves in the cochlea cause the basilar membrane inside to vibrate. As it vibrates, it pushes on so-called hair cells, which have an array of short extensions like small hairs. Pushing on the hair cells thrusts them against a stiff membrane called the tectorial membrane so that the hairs bend. The hair cells are connected to a neuron or nerve cell, so they translate this bending motion into a signal that goes to your brain.
The basilar membrane gradually becomes thinner farther into the cochlea. Different regions of the basilar membrane respond to different pitches. High-pitched sounds like a soprano opera singer cause the thicker regions of the basilar membrane to vibrate, while low-pitched sounds like a bass player generate vibrations in the thinner regions. That's how your brain can tell different pitches apart -- the signals for sounds of different pitches arrive in your brain by different routes.
If you started at the oval window and travelled all the way through the fluid-filled tube in your cochlea, you'd eventually come to another membrane called the round window. The round window is important because the fluid is just water with different chemicals dissolved in it, and water is almost incompressible, which means you can't change its volume by exerting pressure on it. For pressure waves to travel through the fluid, then, the pressure needs to be relieved, and that's what the round window does. By vibrating, the round window membrane relieves the pressure created by the waves generated by the oval window. The oval window and the round window are a little like two sheets of cellophane at opposite ends of a fluid-filled tube -- when one of them vibrates, the pressure it creates will ultimately be relieved by the other.