Interpreting Curl in Vector Fields: ∇×v

[Titan97]

In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl.
Curl{v}=∇×v
But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the gradient points along the direction of maximum increase. But what's the direction of gradient in a vector field? And why does the cross product give the Curl?

 

[Orodruin]

It is not a cross product, it simply happens to have a similar form in Cartesian coordinates.

You can think of (the inner product of) the curl (with a normal to the plane) as being a measure of the line integral around a small planar loop. The direction of the curl is the direction which will maximise this line integral.

This is analogous to how you may see divergence as a measure of the net flow out of the closed surface surrounding a small volume.