A force can change translational motion, but its ability to produce rotation also depends on where and in what direction it is applied. This rotational effect is measured by torque.

About a chosen origin or axis,

$$\boldsymbol\tau=\mathbf r\times\mathbf F,$$

where $\mathbf r$ points from the reference point to the point where the force is applied.

Magnitude

The torque magnitude is

$$\tau=rF\sin\theta,$$

where $\theta$ is the angle between $\mathbf r$ and $\mathbf F$.

Equivalently,

$$\tau=F r_\perp,$$

where $r_\perp$ is the perpendicular lever arm from the axis to the force's line of action.

Same force, different torque

A force applied farther from the axis generally produces more torque. A force directed straight toward the axis has

$$\sin\theta=0$$

and therefore produces no torque about that axis.

This is why pushing near the edge of a door is more effective than pushing near its hinge.

Direction

Torque is a vector whose direction follows the cross-product orientation. In a fixed-axis problem it is often represented by a sign indicating the tendency to rotate in one direction or the other.

Torque is not another name for force: it is the rotational effect of a force relative to a chosen reference axis.