Vorticity is mathematically defined as the curl of the velocity field and is hence a measure of local rotation of the fluid. This definition makes it a vector quantity. 

Circulation, on the other hand, is a scalar quantity defined as the line integral of the velocity field along a closed contour. 

Using Stoke's theorem, the line integral of the velocity field along the closed path, can be expressed as a surface integral of the curl of the velocity field normal to an arbitrary area bounded by the path. But, as already defined, that curl operation is called vorticity. Hence, circulation can be referred to as flux of vorticity. Conversely, it can also be said that that vorticity at a point is essentially circulation per unit area

The last two statements characterize the two quantities, vorticity and circulation, as microscopic and macroscopic respectively. Both these quantities are essentially a measure of the rotation of the fluid flow. 

An interesting theorem involving these quantities is known as Kelvin's circulation theorem which can explain stuff such as smoke rings : 

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As  far as the physical meaning is concerned, circulation can be thought as  the amount of 'push' one feels while moving along a closed boundary or  path.

Vorticity however has nothing to do with a path, it is  defined at a point and would indicate the rotation in the flow field at  that point. So, if an infinitesimal paddle wheel is imagined in the  flow, it would rotate due to non zero vorticity.