Radian
A radian is a unit of angle measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius.
Formula
\theta \text{ (radians)} = \frac{\text{arc length}}{\text{radius}}; \quad 2\pi \text{ radians} = 360 \text{ degrees}
Definition
A radian is a way to measure angles using the radius of a circle: one radian is the central angle that intercepts an arc equal in length to the radius. Since circumference $= 2\pi r$, a full revolution $= 2\pi$ radians $= 360^\circ$, so $1$ radian $= 180^\circ/\pi \approx 57.296^\circ$. Radian measure is the natural dimensionless angle unit defined by $\text{arc length}/\text{radius}$; it makes calculus identities clean, since the limit as $x \to 0$ of $\sin(x)/x = 1$ holds only in radians, and it is the unique parametrization of $S^1$ that makes the exponential map $e^{ix}$ an isometry.
Example
On a circle with radius $5$ cm, walking $5$ cm along the edge turns you through exactly $1$ radian (about $57.3^\circ$), and a full circle is $2\pi \approx 6.28$ radians. Common conversions include $\pi/6 = 30^\circ$, $\pi/4 = 45^\circ$, $\pi/3 = 60^\circ$, $\pi/2 = 90^\circ$, $\pi = 180^\circ$, and $2\pi = 360^\circ$. The Taylor series $\sin(x) = x - x^3/6 + x^5/120 - \ldots$ requires $x$ in radians; in degrees it would need an extra factor of $\pi/180$ throughout, destroying the elegance of the formula.
Key Insight
There are about $6.28$ radians in a full circle because circumference ($2\pi r$) divided by radius ($r$) $= 2\pi$. In degrees, $d/dx[\sin(x)] = (\pi/180)\cos(x)$, but in radians, $d/dx[\sin(x)] = \cos(x)$, removing the messy conversion factor, which is why all of higher math uses radians. Radian measure is "natural" in that it emerges from the geometry of the circle without any arbitrary convention; the choice of $360$ degrees is historical (the Babylonian base-$60$ system), but radians are dictated by the geometry itself.