As a sound wave enters the ear, the outer ear condenses the wave and increases the power (which affects perceived loudness in dB) of the wave. The wave then causes the eardrum to fluctuate inwards and outwards in response to the changing pressures; a pressure higher than that of the inner ear causes the drum to push inwards; a pressure lower than that of the inner ear causes the eardrum to pull outwards. The eardrum then moves a series of bones (named the "hammer," "anvil," and "stirrup.") which, through leverage, further amplify the sound. The last bone, the "stirrup," is connected to a small membrane called the "oval window" and pulls and pushes it back and forth which sets up the wave that entered the ear in the cochlea.
So, the series as a whole goes as follows: A pressure series enters the ear and causes the eardrum to move. The eardrum takes the energy contained in the pressure and converts it to mechanical energy which is transferred through a series of bone levers. The last bone lever causes the oval window membrane to move which converts the mechanical energy of the motion of the bones back into a series of pressures.
The cochlea itsself is a coiled tube which gets narrower towards the end and is divided in half lengthwise by a basilar membrane. The basilar membrane extends the entire length of the cochlea except for a small opening at the narrow end of the tube which allows the two halves to be connected. The basilar membrane contains many thousand audiocilia hair cells which run the length of the membrane--different cells are activated when different frequencies enter the ear and send electrical nerve signals to the brain which are interpreted as sound.
Because the cochlea is less flexible near the wide end and more flexible near the thin end, the sound waves that enter the ear do not retain their exact amplitude along the entire length of the cochlea; this is what causes different parts of the basilar membrane to be receptive to different frequencies. More information can be found in this link.
