For a linear operator between Banach spaces, compactness is a natural strengthening of continuity (boundedness). A linear operator is compact if it sends the bounded subsets to relatively compact subsets.
Since every relatively compact subspace (in a Banach space, or indeed in any metric space) is bounded, every compact operator is bounded. Instead of checking compactness on all bounded subsets it is sufficient to check it for a ball of one fixed radius: an operator is compact iff it sends the ball of unit radius to a relatively compact set.
See also compact self-adjoint operator.
In the setup of Hilbert spaces, instead of a compact operator, one sometimes says a completely continuous operator. However, in the full generality of Banach spaces, by a completely continuous operator one means slightly less: an operator that maps every weakly convergent? sequence to a norm convergent? sequence. For Hilbert spaces and, more generally, for reflexive Banach spaces, the two notions are equivalent.