In mathematics, a unit circle is a circle with a unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S¹; the generalization to higher dimensions is the unit sphere.

Why is the unit circle interesting to a motion designer?
To me it is interseting to get a better understanding of how you can describe rotation in space whether it be 2 or 3 D. Another reason would be  to get a better understanding how Maya stores the rotation values . The rotation of an object is displayed through the user interface as three rotation values: rotate X, Y and Z. Internally Maya stores the rotaion in the transform matrix as values ranging from -1 to 1. The flash movie below shows how a rotation value from 0 to 360 degrees can be visualized in both radians and as a two dimensional coordinate values.

I saw a video on youtube discussing the unit circle and I got this nice trick to remeber the sin, cos and tan functions.
Soh Cah Toa - that reads :

Sin(angle) = Opposite / Hypotenuse.
Cos(angle) = Adjacent / Hypotenuse.
Tan(angle) = Opposite / Adjacent.