Let be the language of set theory. Let S be a particular set theory, for example the ZFC axioms and let T (possibly the same as S) also be a theory in .
If M is a model for S, and N is a substructure of M such that
- the interpretation of in N is
- N is a model for T
- the domain of N is a transitive class of M
- N contains all ordinals of M
then we say that N is an inner model of T (in M). Usually T will equal (or subsume) S, so that N is a model for S 'inside' the model M.
If only conditions 1 and 2 hold, N is called a standard model. A model N is called transitive when it is standard and condition 3 holds. If the axiom of foundation is not assumed (that is, is not in S) all three of these concepts are given the additional condition that N be well-founded. Hence inner models are transitive, transitive models are standard, and standard models are well-founded.
The assumption that there exists a standard model of ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model called the minimal model contained in all standard models. The minimal model contains no standard model (as it is minimal) but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. This model is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not well founded as a relation in the universe, though it satisfies the axiom of foundation so is "internally" well founded. Being well founded is not an absolute property.) In particular in the minimal model there is a model of ZFC but there is no standard model of ZFC.
Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC + a measurable cardinal). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC (like ZF or KP) as well.
There is a branch of set theory called inner model theory that studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact consistency strength of many important set theoretical properties.