A pair of angles are complementary if the sum of their measures is 90 degrees.
If the two complementary angles are adjacent (i.e., have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a right angle.
In Euclidean geometry, the two acute angles in a right triangle are complementary, because there are 180° in a triangle and 90° have been accounted for by the right angle.
The adjective "complementary" comes from the Latin complementum, which derives from the fourth form of the Latin verb complere, i.e. completum for "having been filled up". An acute angle is filled up by its complement to form a right angle, which is the most basic kind of angle, because it can be constructed by drawing the angle bisector of a line. A right angle is a complete (full) angle.
See also: supplementary angles.