The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
For $\mathbf{F}=(u,v,w)$:
$$ \nabla \times \mathbf{F} = \left| \begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial }{\partial y} & \frac{\partial}{\partial z} \\
u & v & w
\end{array} \right| = \left( \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) \mathbf{i}
+ \left( \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) \mathbf{j}
+ \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \mathbf{k}
$$
The curl gives the infinitesimal area density of the circulation of the vector field.
