Definition:
Curl is a differential operator on vector fields in R3. The curl of a field f(x) is denoted curl f or ∇ x f, and it is a vector density field (which is the same thing as a vector field for most purposes). In cartesian coordinates its components are
∇ x f = (∂fz/∂y - ∂fy/∂z, ∂fz/∂x - ∂fx/∂z, ∂fx/∂y - ∂fy/∂x)
The notation ∇ x f comes from the fact that if we consider ∇ = (∂/∂x, ∂/∂y, ∂/∂z) to be a vector then the expression above is that for the vector product of ∇ and f.
Interpretation:
curl f in some sense measures the "rotatingness" of the field f. In particular ∇ x (w x x) = 2w, with the interpretation that the curl of a rigid rotation with angular velocity w is 2w.
Another way to see that curl is related to rotation is to consider the circulation round some small square. If we for simplicity let the opposite vertices of the square be (0, 0, 0), (dx, dy, 0) then the circulation is to linear order
fx(0, 0, 0)dx + fy(dx, 0, 0)dy - fx(0, dy, 0)dx - fy(0, 0, 0)dx = (∂fx/∂y)dydx - (∂fy/∂x)dxdy = (∇ x f)zdxdy
More generally the circulation round the boundary of a small square with area dS and normal n is (∇ x f).ndS. Stokes' theorem is the formalisation of this statement.
A field with ∇ x f ≡ 0 is called irrotational.
Equations:
The following identities hold for scalar fields k, vector fields f, g:
∇ x (kf) = (∇k) x f + k(∇ x f)
∇ x (f x g) = f(∇.g) - g(∇.f) + (g.∇)f - (f.∇)g
∇ x (∇ x f) = ∇(∇.f) - ∇2f
∇ x (∇k) = 0
It is important to remember that the above definition of curl in component form only works in a cartesian coordinate frame. In order to obtain similar expressions for other coordinate frames you have to make some rather messy transformations. For cylindrical coordinates (r, θ, z) and spherical coordinates (r, θ, φ) the respective expressions are
∇ x f = (r -1∂fz/∂θ - ∂fθ/∂z, ∂fr/∂z - ∂fz/∂r, r -1∂(r fθ)/∂r - r -1∂fr/∂θ)
∇ x f = ((r sinθ)-1(∂(sinθfφ)/∂θ - ∂fθ/∂φ), (r sinθ)-1∂fr/∂φ - r -1∂(r fφ)/∂r, r -1(∂(r fθ)/∂r - ∂fr/∂θ))
The expressions may be hard to decipher, but I assure you that it is more painful to derive them yourself.