The nontrivial standard examples of strongly minimal theories are the one-sorted theories of infinite-dimensional vector spaces, and the theories ACFp of algebraically closed fields of characteristic p. As the example ACFp shows, the parametrically definable subsets of the square of the domain of a minimal structure can be relatively complicated ("curves").
More generally, a subset of a structure that is defined as the set of realizations of a formula φ(x) is called a minimal set if every parametrically definable subset of it is either finite or cofinite.
It is called a strongly minimal set if this is true even in all elementary extensions.
A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid.
Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem.