The sample rate determines the highest frequency that can be reproduced (Google "Nyquist Theorum" for more). Human hearing is limited to around 20Hz to 20,000Hz. A 44.1kHz sampling rate (like a CD) will accurately reproduce 22,050Hz - Slightly higher than the basica human hearing range, and considerably higher than a lot of audio equipment.
Got that so far?
Bit depth (a.k.a. word length) is measured in bits and is a measurment of the resolution OF the sample. A 1 bit sample has only two possible values - "ON" and "OFF." When you go up, the numbers go up exponentially - On a 16 bit recording (like a CD), any one sample (there are 44,100 of them per second) can have one of around 65,000 values. That's a reasonable resolution. A 24-bit signal on the other hand will have one of 16.7 million values - A huge jump in accuracy, and basically the "standard" word length for audio recording.
So what you find the majority of the time - Rookies, Hobbyists, Semi-professionals, Recording Top Guns - is 24-bit recording *at the target sample rate* (44.1kHz for music CD production, 48kHz for video, etc.). Some prefer to record at higher sample rates, others don't. There are sensible arguments on both sides. But one thing that few (if anyone) will deny is that if you can't get 44.1kHz to sound absolutely wonderful, going up to a higher sample rate isn't really going to help.