An improper rotation can be regarded as the combination of an ordinary rotation of real 3-dimensional space R3 that keeps the origin fixed, followed by an inversion (x goes to -x). An improper rotation of an object thus produces a rotation of its mirror image. Improper rotations are described by 3-by-3 orthogonal matrices with a determinant of -1.
A proper rotation is simply an ordinary rotation, which has a determinant of 1.
The product (composition) of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is a improper rotation.
When studying the symmetry of a physical system under an improper rotation (e.g. if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general; between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).
See also: Isometry