maximal ideal
Among all the proper ideals of a ring, a maximal ideal is one that is as big as it can be without becoming the whole ring — you cannot squeeze any larger proper ideal between it and R. It sits right at the ceiling of the lattice of proper ideals.
Precisely, a proper ideal M of a commutative ring R with identity is maximal if the only ideals containing M are M itself and R. The defining algebraic payoff is that R/M is a field: collapsing a maximal ideal to zero produces the cleanest possible quotient, one in which every nonzero element is invertible. This is the source of the maxim 'maximal ideals are the points where you can evaluate a function and get a number.'
Maximal ideals always exist in a ring with 1: by Zorn's lemma every proper ideal is contained in some maximal ideal. They are exactly the closed points of Spec(R), and in the Nullstellensatz of algebraic geometry the maximal ideals of a polynomial ring over an algebraically closed field correspond bijectively to points of affine space.
In Z[x] the ideal (2, x) is maximal: Z[x]/(2, x) ≅ Z/2Z, a field with two elements. By contrast (x) alone is only prime, since Z[x]/(x) = Z is a domain but not a field.
A maximal ideal of Z[x] and the two-element field it produces.
In a noncommutative ring one distinguishes maximal left, right, and two-sided ideals; the Jacobson radical is built from maximal left ideals. In the commutative world these distinctions vanish.