This is the lowest order monic polynomial satisfied by an object, such as a matrix or an algebraic element over a field.

For instance, $\sqrt2$ is an algebraic number, that is, it's a root of a non-zero polynomial with rational coefficients. Its minimal polynomial is $x^2-2$, since $\sqrt2$ is a root of this (monic) polynomial, and it is *not* a root of a non-zero polynomial with rational coefficients of a smaller degree.