What is it?
A signal with no frequencies above is completely determined by samples taken at more than per second, and can be rebuilt exactly with sinc interpolation. Sample slower and high frequencies masquerade as low ones (aliasing). That is why CDs use 44.1 kHz for 20 kHz hearing.
Formulas
The mathematics behind it
Sampling replicates the spectrum; Nyquist's condition keeps the copies from overlapping.
Where is it used?
Computing topics reachable from here, through the chain of ideas that leads to them:
What depends on it
This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.