Torque, a measure of "twist" on a shaft, is as basic as pressure and stress in engineering mechanics. It differs from a bending moment because the vector sum of forces for torque is zero. The equation is simple, but if you are trying to calculate torque experimentally, you will need some measuring devices to obtain the values necessary in the equation. At a minimum, you need to know the amount of force applied to the system and the distance from the shaft to the point where the force is applied.
Things You'll Need
 Protractor
 Pressure gauge
 Strain gauge
 Spring scale
 Other force or angle measuring devices

Measure or otherwise determine the distance (d) from the center of the shaft to the point of force application. Use units consistent with standard torque units, which are "ftlb" for English units or "Nm" for metric, or SI, units.
Example:
d = 0.5m (measured) 
Measure or otherwise determine the magnitude of the force (F) applied to the system. If necessary, convert to "lbs" or "N," as applicable. If the force is in vector form, you can calculate the magnitude using the vector dot product.
Example:
F = (4i + 2j  4k) N
F = SQRT(44 + 22 +(4)*(4))
F = SQRT(16 + 4 + 16) = 6 N 
Measure or otherwise determine the angle (a) at which the force is acting. If the force is acting tangent to a circle with radius "d" and with its center at the center of the shaft, then the angle is 90 degrees. Calculate the amount of the force contributing to the torque (Ft).
Example:
a = 45 degrees
Ft = Fsin(a) = 6sin(45) = 4.242 N 
Calculate the torque by multiplying d and Ft.
Example:
T = dFt = 0.54.242 = 2.121 Nm
Tips & Warnings
 Motors use a different formula for torque than mechanical systems. Torque for motors uses horsepower and RPMs, both of which must be determined by measurement or by some other means. For motors, find torque (in ftlb) using this equation: T = (HP*5252)/rpm.
 "Other means" could be manufacturer's specifications, engineering charts, given information in an academic environment or design estimation.
References
 Vector Analysis; Dr. Murray R. Spiegel; 1959
 College of Micronesia: Levers and Torque
 Photo Credit torque wrench and accessories image by Christopher Dodge from Fotolia.com