The measurable subsets of a measure space $(X,\mu)$ are those subsets $A$ of the underlying set $X$ for which the measure $\mu(A)$ is defined (at all, even possibly as infinite). Intuitively, one might expect every subset of $X$ to be measurable, and this is the case in some examples, but in the standard example of Lebesgue measure on the real line, this is incompatible with the axiom of choice. (On the other hand, in dream mathematics, where the full axiom of choice fails, every subset of the real line is Lebesgue measurable.) Regardless, every subset $A$ of $X$ has both an outer measure? $\mu^*(A)$ and an inner measure? $\mu_*(A)$.
The concept actually makes sense in any measurable space as well as in related contexts such as Cheng spaces and measurable locales.
Typically, the notion of measurable subset of $X$ is given axiomatically by a structure on the set $X$, usually a $\sigma$-algebra $\mathcal{M}$. The a subset of $X$ is measurable if it belongs to $\mathcal{M}$.
Sometimes $\mathcal{M}$ is a weaker structure, such as a $\delta$-ring; see other variants at sigma-algebra. Then we require some subsidiary notions: $S$ is relatively measurable if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is $\sigma$-measurable if it is a union of a countable family of elements of $\mathcal{M}$.
(The terms ‘relatively measurable’ and ‘$\sigma$-measurable’ are my own; I cannot find them in the literature. In the case of $\sigma$-measurable sets, the terminology follows a standard pattern. Halmos uses relatively measurable sets, but he doesn't seem to give them a name.)
Besides the $\sigma$-algebra of measurable subsets, we may place another structure on $X$, a $\sigma$-ideal $\mathcal{N}$ in $\mathcal{M}$. (This structure also exists, for example, for any measure space, and already for a Cheng space or a localisable measurable space.) Then a null set is any subset of $X$ (measurable or not) contained in an element of $\mathcal{N}$. The null sets form a $\sigma$-ideal $\bar{\mathcal{N}}$ of the power set $\mathcal{P}X$, and we may equivalently begin with the $\sigma$-ideal of null sets as long as every null set is contained in a measurable null set.
This allows two complementary modifications to the notion of measurable set:
We may accept the union of any measurable set and any null set as measurable. Since this changes the meaning of ‘measurable’, we may speak of $\mathcal{M}$-measurable and $(\mathcal{M},\mathcal{N})$-measurable sets. The collection of such sets may be denoted $\mathcal{M} \cup \bar{\mathcal{N}}$ (applying $\cup$ pointwise).
We may regard two measurable sets as equivalent if their symmetric difference is a null set (and hence an element of $\mathcal{N}$). This defines an equivalence relation on $\mathcal{M}$; the collection of equivalence classes is denoted $\mathcal{M}/\mathcal{N}$.
Note that $(\mathcal{M} \cup \bar{\mathcal{N}})/\mathcal{N} \cong \mathcal{M}/\mathcal{N}$; indeed, this diagram commutes:
Accordingly, one may skip the former modification if one intends to also perform the latter. Nevertheless, even when using $\mathcal{M}/\mathcal{N}$ as the lattice of measurable ‘sets’, if one considers a subset $A$ of $X$ and asks whether $A$ is ‘measurable’, one usually means whether $A \in \mathcal{M} \cup \bar{\mathcal{N}}$.
One could equally well begin with a $\delta$-filter $\mathcal{F}$, although a $\sigma$-ideal is more traditional. Then a full set is any subset of $X$ that contains in an element of $\mathcal{F}$. (If we start with the $\delta$-filter $\bar{\mathcal{F}}$ in $\mathcal{P}X$, then every full set must contain a measurable full set.) In constructive mathematics, full sets are more fundamental for such examples as Lebesgue measure. In any case, the modifications are as follows:
We may accept the intersection of any measurable set and any full set as measurable; the collection of such sets may be denoted $\mathcal{M} \cap \bar{\mathcal{F}}$.
We may regard two measurable sets as equivalent if their biconditional is a full set; the collection of equivalence classes is denoted $\mathcal{M}/\mathcal{F}$.
Then we have this commuting diagram:
Already we have seen that we may be more interested in equivalence classes of measurable sets than in the sets themselves. We may well start with any appropriate algebra in the place of $\mathcal{M}/\mathcal{F}$ above and regard its elements are ‘measurable sets’. This may be done in the theory of measurable locales and other ‘pointless’ approaches to measure theory.