Explanation about Curl in electromagnetism what does curl means in electromagnetism? How can we derive relation between curl and gradient in electromagnetism?
 A: As stated in Wolfram's Mathworld (http://mathworld.wolfram.com/Curl.html) and many treatments on vector calculus (Arfken and Weber's, Morse and Feshbach's books, for example), the component in direction $\mathbf{n}$ of the curl of a vector field $\mathbf{F}$ can be defined as
$$
(\nabla\times \mathbf{F})
\cdot\mathbf{\hat{n}} = \lim_{A\to 0} \frac{\int_C\mathbf{F}\cdot d\mathbf{l}}{A}$$
where $d\mathbf{l}$ is a differential displacement along $C$, $C$ is the contour delimiting $A$, and $A$ is some area centered at the position $\mathbf{x}$ of the vector field $\mathbf{F}$.
The curl operator is important in electromagnetism because it allows us to write both Faraday's and Ampere's law in differential form
$$
\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}.
$$
This expression is more powerful than the integral one because it implies (by Stokes's theorem) that the integral formulation of Faraday's law holds for any surface drawn in space. By the curl definition, you can think of the differential form implying that Faraday's integral law is satisfied by any infinitesimaly small area in space and, thus, by any sum of such areas.
There is no general relation between the curl and gradient operators. The curl operator takes a rank $n \geq 1$ tensor and returns a tensor of the same rank (i.e. acts on vectors and returns vectors) and the other increases the rank of the tensor on which it acts by one (i.e. turns a scalar into a vector). You could also see that they are generally different using Einstein's notation.
A: Curl is not unique to EM. It's a part of the vector math that you should know before attempting to do EM with Maxwell's equations. It may help you to look at an image example that I grabbed off of Google:

The best explanation that I can give of the curl of a vector field (such as the electric or magnetic field) is a curling motion as is shown in the example. I believe in Griffith's, this is explained as leaves in a stream.
When it comes down to it, you should really think of curl as its mathematical representation -- you shouldn't trust your intuition with things like this:
$$
\nabla \times \vec{A} =
\begin{vmatrix}
\hat{x} & \hat{y} & \hat{z} \\
\partial_x & \partial_y & \partial_z \\
A_x & A_y & A_z
\end{vmatrix}
$$
One thing to note is that if there is no cross-dependence of the components on the other coordinates, the curl will be 0. If there is cross-dependence, you should work it out.
If you want a resource, the first couple of chapters of Griffith's Introduction to Electrodynamics should be rather short and hopefully helpful. It offers worked examples throughout and the first chapter is dedicated to vector math.
