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.
Here is a diagram showing the relation between these:
A general axiomatic framework for large cardinal axioms is proposed in
Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces: towards a unified theory of large cardinal and embedding axioms, Annals of Pure and Applied Logic Volume 41, Issue 2, 6 February 1989, Pages 93–106
Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces. II. Limit ultraproducts and iterated embeddings, Acta Cient. Venezolana 40 (1989), no. 5-6, 311–318.