16 bit means that when you convert an analog signal to digital you are using a 16 digit binary word to represent each sample. A 16 bit binary word can take on 2**16 = 65,536 different values. If you are using 24 bit then each binary word can take on 2**24 = 16,777,216 different values.
Imagine that the maximum amplitude signal your system can accept is 10V and that you are inputting a 10V sine wave for analog to digital conversion. A 16 bit system will produce a digital output where the smallest difference between two digital samples is 10/65,536 = 0.000152 Volts. This is sometimes called the granularity of the system. A 24 bit system would have a granulaity of 10/16,777,216 = 0.000000596. The 24 bit system produces a much smoother and more accurate digital representation of the signal. This becomes more important when you are converting signals with a lot of high frequency content.
The difference between 24 and 16 bit systems also affects the dynamic range of the signal once it's converted back to analog. Dynamic range is the ratio of the loudest signal to the quietest signal a given bit width can produce. I forget the formula...
44.1 Khz is the sampling rate and means that the system is sampling/converting the signal 44,100 times a second. Again this becomes more important at higher frequencies.
It's not to hard to get people here arguing whether 24 bit resolution is noticeably better than 16, or whether 96Khz sampling rates are really necessary. Ultimately it comes down to deciding how much resolution you need and whether your ears can tell the difference. I sample at 44.1 Khz/24 bits because...I can.
