A vortex is formed in a flow, where there is a distribution of the Re-number over the cross-section area of the flow. With some obstacle across the flow vortex could be generated at low Re numbers and the frequency of vortices is used to measure fluid velocity. Search the web on vortex flow meter or get a book on basic fluid dynamics.
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A free vortex is one caused by the angular momentum in the fluid. For example, when you drain the sink and a vortex forms it is a free vortex where the principle of conservation of angular momontum causes the the fluid to speed-up(higher rpm) as it approaches the drain. Although it would appear that the water in the sink was not moving before the plug was pulled, even a small movement in the water or perturbation in the flow toward the drain will result in a vortex.
An example of a forced vortex is a propeller in a tank. The whole tank will turn at the same rpm except for the effects of wall drag.
By noticing the differences between both vortices one can easily imagine how they are formed.
In free vortices the rotating speeds are inversely proportional to the radii of the vortex circles, meaning highest velocities are at the center, like the water swirl in a drain, or a tornado.
In forced vortex it is just the opposite, the speeds are proportional to the radii, and the maximum speeds are at the periphery, approximated by the rotation of a rigid body.
If we put a glass with water on a turntable, by compelling the whole water body to rotate, we get a forced vortex.
