It was introduced by J. Buhler and Z. Reichstein[1] and in its most generality defined by A.
[2] Basically, essential dimension measures the complexity of algebraic structures via their fields of definition.
For example, a quadratic form q : V → K over a field K, where V is a K-vector space, is said to be defined over a subfield L of K if there exists a K-basis e1,...,en of V such that q can be expressed in the form
with all coefficients aij belonging to L. If K has characteristic different from 2, every quadratic form is diagonalizable.
Technically, one always works over a (fixed) base field k and the fields K and L in consideration are supposed to contain k. The essential dimension of q is then defined as the least transcendence degree over k of a subfield L of K over which q is defined.