Just as the class P can be thought of as the tractable problems, so NC can be thought of as the problems that can be efficiently solved on a parallel computer. NC is a subset of P because parallel computers can be simulated by sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are some tractable problems which are "inherently sequential" and cannot significantly be sped up by using parallelism. Just as the class NP-Complete can be thought of as "probably intractable", so the class P-Complete can be thought of as "probably not parallelizable" or "probably inherently sequential".
The parallel computer in the definition can be assumed to be a parallel, random-access machine (PRAM). That is a parallel computer with a central pool of memory, and any processor can access any bit of memory in constant time. The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor. It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models.
Equivalently, NC can be defined as those decision problems decidable by uniform Boolean circuits with polylogarithmic depth and a polynomial number of gates.
Stephen Cook coined the name NC ("Nick's class") after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.