What determines the sharpness of a shadow? What are the factors that affect the sharpness of a shadow? 
I would think that the distance between the light source and the object, the distance between the object and the shadow, and the size of the light source would all have an effect. 
How do they affect the shadow exactly? What is the full explanation?
 A: If you are talking about the partially shaded outer region of a shadow then yes it would be called the penumbra and would be described as Jen says. Or like the area of a partial eclipse. 
If your talking about factors that affect the sharpness of a shadows edge then it would have to do with distance, wavelength, light source and sharpness of the illuminated  objects edge.
Light is being diffracted by the edge of the object being illuminated. This is called knife edge or straight edge diffraction and it produces a diminishing fringe pattern along the shadows edge that looks like this.

If the edge is sharp enough (like a razor edge) and the light source is monochromatic you can see these fringe patterns. The term I use is single edge and I write about it in my paper on the web site at the top of my page. 
A: The finer light source smaller than the object casting the shadow will make the most defined shadow.
Figure adapted/modified from "Shadows" via School physics by Keith Gibbs

A: Although the Sun is much wider than the Earth it produces sharp shadows on a clear day since its angular width from Earth is very small.  The slightest mist reduces the sharpness of the shadow.
At night a bare light-bulb produces a much sharper shadow than the same bulb inside a diffuser, such as a Chinese lantern. 
