measurable set
A measurable set is simply a set you are allowed to measure: one that belongs to the chosen sigma-algebra. Think of the sigma-algebra as a guest list; the measurable sets are the sets on the list, the ones the measure mu is willing to assign a size to.
Formally, given a measure space (X, A, mu), a set E is measurable (more precisely, A-measurable) if E is an element of A. Membership in A is the entire definition — measurability is a property of belonging, decided before any value is computed. Once E is measurable, mu(E) is a well-defined element of [0, +infinity].
Which sets count as measurable depends entirely on which sigma-algebra you fixed. For Lebesgue measure on R one uses the Lebesgue sigma-algebra, built from the open sets together with all subsets of null sets; it contains every Borel set and strictly more besides. The honest caveat is that not every subset of R is Lebesgue measurable — non-measurable sets exist, though constructing one requires the axiom of choice and none can be written down explicitly.
Every interval, every open set, every closed set, and every countable set in R is Lebesgue measurable. The set of rationals Q is measurable with measure 0; the Cantor set is measurable with measure 0; a Vitali set is not measurable at all.
Almost every set one writes down is measurable; the exceptions must be summoned by choice.