One radian is 0.1591549431 turns, since a full turn is 2π radians. The turn is the most intuitive angle unit there is — one turn is once around — and it is quietly gaining ground in teaching and in software.
About the units
A turn, also called a revolution or a cycle, is one complete rotation. It is the unit rotational speed is actually measured in: revolutions per minute is turns per minute, and hertz is turns per second. Despite that, mathematics has historically expressed angles in radians and multiples of π, which means a full circle is 2π and a quarter turn is π/2 — expressions that beginners consistently find harder than they need to be. The proposal to use τ = 2π as the fundamental circle constant, which would make a full turn τ radians and a quarter turn τ/4, is an argument about exactly this.
The exact factor
The relationship is 1 turn = 2π radians = 6.283185307179586 radians, so 1 radian = 1/(2π) = 0.15915494309189535 turns. Some equivalents: a quarter turn is π/2 radians or 90°, a half turn is π radians or 180°, and a full turn is 2π radians or 360°. The connection to rotational frequency is direct and worth stating: an object spinning at 1 Hz completes one turn per second, which is 2π radians per second — the angular velocity ω that appears in every oscillation equation.
Where you meet this conversion
Rotational mechanics, where angular velocity in radians per second and rotational speed in revolutions per minute describe the same motion and differ by a factor of 2π/60. Motor specifications give rpm, control systems compute in rad/s. In computer graphics some libraries take turns rather than radians or degrees precisely because it removes the conversion — a rotation of 0.25 is unambiguous. Gear ratios, wheel revolutions and cyclic phenomena from tides to alternating current all count naturally in turns.