A large cardinal is a cardinal number that is larger than can be proven to exist in the ambient set theory, usually ZF or ZFC. Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength of various other assertions that are unprovable from ZFC.
axiom of infinity – a large cardinal axiom relative to finitist theories, but not relative to ZF
inaccessible cardinal – the smallest sort of large cardinal in ZF, equivalent to the existence of a Grothendieck universe.
measurable cardinal – the boundary between “small” large cardinals and “large” large cardinals
elementary embedding – a tool used in the study of large large cardinals
Vopěnka's principle – a large cardinal axiom with important implications for the behavior of locally presentable categories and accessible categories.
Wikipedia has a list of large cardinal properties.