Suppose you need to determine how much torque is required to lift a load, cause a wheel to accelerate or to make a conveyor belt move. If you know how much force is required at one radius (arm length) of leverage, you can easily convert the torque requirement for another arm length. The relevant equation is Torque = Perpendicular Force x Radius about the centre of rotation.
Draw a diagram of a pulley wheel of radius R with a mass m hanging off of it. You can translate this example to a range of torque problems, where the load applies a perpendicular force at radius R from the centre of rotation.
Determine the force created by the mass. In this case, use Newton's second law to get F=ma=mg, where g is the gravitational acceleration constant, 9.80 meters per second squared.
Calculate the torque you'll need to apply to the pulley to keep the weight from dropping. In other words, FR = mgR is the torque needed. So if you use a motor to drive a wheel of radius r attached to the same axle as the pulley, then the motor needs to apply a force of F = mgR/r.
Note that when you apply torque equal to the torque that you're opposing, as in the pulley example, this is only enough torque to maintain equilibrium. To actually move the load you're opposing, you need to exceed this torque by a small amount.