Models in set theory and continuum hypothesis

To construct a model of set theory means to produce a set $A$ and a relation $R$ on $A \times A$ such that all the axioms of ZFC are satisfied if we take "set" to mean "element of $A$" and take "$a \in b"$ to mean $aRb$.

This is not actually any different than the case with groups. The signature is different, and the axioms are different, but the definition of "model" is the same.

There is one complication, though. Although ZFC proves that there is a model of the group axioms, ZFC does not prove that there is a model of the ZFC axioms. One way we can get around this is by moving to a stronger system of set theory to construct the model of ZFC. For example, Kelley-Morse set theory proves that there is a model of ZFC. Another way is to simply assume there is one model, and use that to construct other models.

Sometimes, in set theory, we allow a more general kind of model in which $A$ is a proper class and $R$ is a definable relation on pairs of elements of $A$. These are called "class models".