Bits refer to binary. Binary is a base-2 mathematic system, meaning that only a one or a zero can be used. So, 16-bit means that there are sixteen digits, all ones or zeroes; 24-bit means that there are twenty-four digits, all ones or zeroes; and so on. So the bit value for 16-bit would equal 216 or 65,536. This means that each sample recorded at 16-bit can contain any one of 65,536 unique values made up of 16 ones and zeroes.
All this sounds terribly complex (and is) but what does this mean in the real world? Each sample of audio contains data that provides a digital-to-audio converter with the information necessary to accurately reproduce the analog sound wave. Things like dynamic range, frequencies, etc are all contained within this data stream. If the bit depth is too low, data will be lost in each sample by stripping away a little bit of every piece of data and the reproduced sample will be degraded. Conversely, the higher the bit depth, the more data will be captured to recreate the sound.