How should I understand curl? I understand that curl, in a simplified way I suppose, measures the magnitude of vorticity at any point.  Suppose I have the function provided by wikipedia, $$\vec{F}(x,y,z) = -x^2\hat{y}$$  The vector field looks as follows

with corresponding curl

My confusion arises when evaluating $\nabla \times F = -2x\hat{z}$ at a point.  At x = 1, the function has a curl of -2.  What does this mean?  I understand that if I were to put a paddle wheel near x=1, the wheel would experience a torque and would turn, assuming the vector field models fluid.  What I don't understand is how we can quantify the curl at a point.  If I were to place a "point-sized" paddle wheel at this point, it would simply translate, not rotate.  
What am I missing? 
 A: The infinitesimally tiny paddle (but having a tiny surface where the fluid can exert surface forces) would translate and rotate about its own axis as well. The curl relates to how fluid particles in a tiny neighborhood of the point move relative to each other. In the limit of a infinitesimally small neighborhood, that neighborhood rotates about the point in question with an angular velocity which corresponds to the curl as below. 
The angular velocity corresponding to that rotation would be half the value of the curl that you just computed (called the  vorticity of the vector field in the context of fluids).  The tinier the paddle, the closer its angular velocity is to half the magnitude of the curl. 
This exercise you have done is helpful in getting yourself some intuition regarding the curl. Do check out the examples in the link for more of a feel for it. There are some properties and useful theorems also on that wiki page. 
Presence of rotation/vorticity does not mean the flow streamlines cannot be straight.  Try placing a small paddle near the edge of a stream (anywhere except the exact middle of the channel) with parallel flow (like a drain with parallel sides). As in this question, the streamlines are not curved. Yet, watch the paddle translate and rotate.  Or you can see that happen at 3:00 in this video.  The paddle (“vorticity meter”) is described in the preceding minute. Likewise, simply having curved streamlines does not mean there is vorticity. An example is the third vector field illustrated in the Wikipedia link. Streamlines are circular but a paddle placed there would translate, but not rotate at all. 
A: Curl, like divergence, is the result of a limit. A geometric point will feel the same force on both its sides and will not rotate, only translate.
However, curl and divergence are great for calculating the observable/relevant quantities of circulation and flux, respectively, as pointed out in Stoke's theorem.
