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It definately affects the sound quality.
I'm not going to give the full explanation about samplerate and bit depth, cuz that will take very long, but I'll try to simplify as much as possible.
The signal that comes out of your mixer (well most actual mixers anyway) is analog (a certain voltage). Computer data, thus also audio, is digital data (1 and 0's). So when you are recording to your computer, the analog signal will be converted to digital data (your soundcard contains elements called analog to digital convertors at the inputs, and digital to analog convertors at the outputs).
But what is the main difference between analog and digital. Well, an analog signal is a continuous signal (you have a certain value for each moment), while digital data is discrete (you only have a certain value for each time segment). So to convert analog to digital, we're gonna do what's called sampling. We are going to take a value each X time. And that X time is defined by the sampling frequency. Hertz is the opposite of second (1 hertz = 1/1 second). So for example if you say the sampling frequency is 44100 Hz, a digital sample will be taken every 1/44100 second. Or in other words, in one second 44100 samples will be taken.
Why are there different sampling frequencies, and why are they always funny numbers. That's because there's a certain law, from Shannon-Nyquist. That law says that the sampling frequency must be at least double the value of the highest frequency that's present in the analog signal you are sampling. If this requirement is not met, you'll get aliasing. I'm not really going to explain that further, just know, aliasing isn't good. You'll hear artifacts.
The most used frequency nowadays (cd for example) is 44100. It is said that the normal bandwith of an analog audio signal is 20 Hz to 20 KHz. So the highest possible frequency should be 20 KHz. When we apply Nyquist, the sampling frequency should be at least 40 KHz, but they left a margin (because at the end there's a low pass filter against aliasing, and it hasn't an infinitely steep cut), so they set it to 44.1 KHz.
Ok, but why are there lower or higher sampling frequencies also? Well, lower that's if you need more space, or if a high frequency isn't required.
But why higher? It takes up more space (more samples per second). Again this is because of that aliasing problem. I told you something about a low pass filter didn't I? Well there's a problem with analog filters. If they have a steep cutting curve, there will be problems concerning phase. Phase problems are bad (you lose some frequencies and such). When using higher samplerates the filter has to be less steep. Less steep, less phase problems. Easy as that. On some occasions (cdplayers for examples) they even use oversampling. Sampling at very high rates, so they can add digital filters (using delays off all sorts), which gives an even better result).
Now, you must know most consumer cards (Sounblaster and such) can record up to 48000 Hz. 48 kHz was (and still is in most studio's) the standard sampling frequency (it's shifting to 96 kHz now). But for home recordings, 44.1 kHz is the best way to go, as most of the time it's for burning on cd. And cd only accepts 44.1 KHz 16 bits. Most people aren't familiar with downsampling and dithering on professional level, so stick with the standard rates. Besides, higher frequency means more hard disk space...
Phew, that was for sampling frequency, next chapter : bit depth.
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