The radian is one choice for the unit of measure of an angle. The angle formed by an arc enclosing a point (thus, a circle and its center) is 2 pi radians, where pi is the ratio of the measure of the circumference of the circle to the measure of its diameter. From the election of a circle corresponding to 360 degrees of angle, one radian of angle is equal to 57.3 degrees. The measure of arc length corresponding to one radian of angle is equal to the measure of the radius being used to create the arc.
The radian gets identified from Calculus as the "natural" measure of angle such that the action of calculus operators on the trig-functions is cleaner.