What pitch range can the human ear hear?

The best one line answer is: 20 to 20 000 Hz.

A better answer:
The basilar membrane is a cord of varying thickness and width that sits in the inner ear. Different parts of it vibrate in response to different frequencies from the outside world. (There's a wonderfully clear video of this here). This pretty much sets the limits on what frequencies the human ear can hear. If the basilar membrane can't vibrate at the same frequency of the sound entering the ear, you won't hear it.

When things vibrate, they oscillate (move up and down). 'Frequency' is the measure of the number of full oscillations (up to down and back up again) that take place in a second. The higher the number of oscillations, the higher the frequency. Frequency is measured in Hertz (Hz).

So, sounds with a frequency of 20 Hertz will set most of our basilar membranes in motion, but sounds with a frequency of 15 won't. The upper limit of most young people is about 20 000 Hz, but the higher frequencies are generally lost with age, so that in older people the range is reduced. This fact was exploited by some town councils: they broadcast irritating high-pitched sounds around areas where they didn't want teenagers to loiter. To an adult, the sounds were simply inaudible, but to a teenager (or, presumably, a sleeping baby) the noise was apparently extremely unpleasant. (Glad to say, this idea seems to have been largely binned.)

So sounds above our upper limit, as with sounds below our lower limit, simply aren't audible. But I hope I've emphasised that this is a purely human constraint. For instance, much of whales' 'songs' are lost to us since they communicate within an different audible range - one that goes much lower down. Similarly, dogs (for instance) can hear much higher frequencies than we can. That's the principle behind dog whistles - we can't hear them, but dogs can - loudly.