What is a radian?
An angle of 1 radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length 1. The picture below illustrates a the radian definition.
To explain what a radian is to people that have never heard of it I would say look at a unit circle, like the one pictured bellow, and in between every dot you see is a radian. It's that simple.
How do unit circles and radians relate?
A unit circle and radians relate because one radian is equal to the radius of the circle. And as you move around the circle you get more radians. Using the unit circle allows us to switch radians to degrees and degrees to radians.
How does the formula for circumference of a circle and radian measure relate?
The formula for circumference is 2(pi)r which is 2pi radians.
How do radians relate to degrees? Which seems more mathematically "pure"?
A circle is 360 degrees which is the same thing as 2pi radians, or if you go around the circle 180 degrees it is the same as 1pi radian. I prefer degrees because they are much easier to work with, and to understand. When you go around a circle 120 degrees you know right away that it's 120 degrees, if you have 2(pi)/3 then you have to go through the trouble of converting it to degrees. Radians is more than likely the most "pure" out of the two of them, just because it's more complex.