Sample rate conversion is the process of converting a digital signal from one sampling rate to another, while changing the information carried by the signal as little as possible.
Source: Wikipedia
SRC is often used in DACs. The reasons are explained here.
In PC audio it is needed when your soundcard supports 48 kHz only and you play a (ripped) CD (44.1 kHz).
The mathematics are complex. If you can read this ( Shannon's sampling theorem ):
You might try The Digital Audio Resampling Home Page'', by Julius O. Smith III.
Asynchronous SRC can provide very good jitter rejection but that it will add some distortion and noise to the signal.
ASRC can be made with extremely high dynamic range and SNR, but will then inevitably be of very high complexity.
The ups and downs of arbitrary sample rate conversion - Ivar Løkken, 12/4-05.
Infinite Wave tested a lot of implementations.
Some examples of a swept sine wave with -6 dbFS peak amplitude, spanning the frequency range from 0 to 48 kHz for 8 seconds

Secret Rabbit Code 0.1.2 (ZOH)

Logic8 (Tiger)

R8brain Free
Obvious SRC implementation has its impact on sound quality.
Benchmark Media provides an example of the impact of sample rate conversion in OSX 10.4.6

A 16-bit 10k sine wave played through iTunes on OSX 10.4.6, without any sample-rate conversion.

A 16-bit 10k sine wave played through iTunes on OSX 10.4.6, with sample-rate conversion from 48 kHz to 44.1 kHz.
The distortion seen in this graph is solely caused by sample-rate converting from 48 kHz to 44.1 kHz. As seen in the graph, the signal-to-noise ratio went from 130 dB to less then 80 dB!
Uli Brueggemann did a nice experiment.
Upsample the original 44.1 to 176 and back to 44.1
Original signal
Ideal upsampling to 176
Ideal downsampling from 176 to 44.1The conversion has been done using a sinc-filter with 65536 taps
The same procedure using Adobe Audition
Original signal
Adobe Audition upsampling to 176
Adobe Audition downsampling from 176 to 44.1
The same procedure using SRC (Secret Rabbit Code) using cPlay
Original signal
Secret Rabbit Code upsampling to 176
Secret Rabbit Code downsampling from 176 to 44.1
Obvious both Audition and Secret Rabbit Code are unable te reconstruct the original signal.
Shows you that sample rate conversion can have a profound impact on sound quality.