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I know how shading works with faces:

Smooth cylinder Flat cylinder

That's set by choosing Smooth/Flat under Faces in the Shading menu:

Shading - Faces: (Smooth/Flat) - Edges: (Smooth/Sharp) - Vertices: (Smooth/Sharp)

But what do the other two options—Edges and Vertices—do?

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This feature makes it possible to sharpen or soften individual components of a mesh instead of soften or sharpen the entire mesh. Especially for Low Poly Modeling it can be useful to have more control about the object's appearance and thus to define a more complex shape with less geometry.

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Result of sharpen the selected edges only:

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Note: In Order to make it work, enable Auto Smooth Option in the Properties Panel > Data.

When sharpening vertices, the connected edges are also affected:

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Render result of sharpen edges only:

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    $\begingroup$ This is very useful if you want to create a Terminator effect... :) $\endgroup$ – David Feb 1 '16 at 19:19

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