Shadow Catcher

To be clear upfront: The Shadow Catcher shader is simply a much easier and faster method of achieving the same effect that can also be achieved using other methods, e.g., with the Compositing tag, multi-pass tricks, etc.

In short, the Shadow Catcher shader integrates shadows and, if desired, reflections into the Alpha channel.

Only shadows and/or reflections from objects to which shaders are assigned will land in the Alpha channel (the remaining space in the Alpha channel will be black).

This makes it easy, for example, to insert rendered objects in photos in a compositing program, i.e., insert them into a two-dimensional background (backplates):

From left to right: The 3D object; the rendered Alpha channel; the 3D element inserted into the photo.

The following conditions must be met:

The Shadow Catcher shader must lie on the object onto which the shadow or reflection is cast (this would be the black plane in the image above; a Floor object would also work just as well)
The Alpha Channel option must be enabled in the render settings’ Save menu
The lighting situation should match that of the background into which the rendered object will be inserted as closely as possible. In the image above, the light that casts the shadow should be positioned behind the figure to the right. Often, a second light source is used that creates a uniform environment light (somthing like Ambient Illumination).
Render the scene. The saved image file can now be opened in a compositing program and inserted into a background image, where shadow and reflection will be displayed according to the Alpha channel’s settings.

Limitations

Note that GI effects cannot be taken into account by the Shadow Catcher with regard to shadows since it only works with shadows from light sources.
The Shadow Catcher cannot exclude itself from self-shadowing. To do so, the Self Shadowing option must be disabled in the Compositing tag.
The Shadow Catcher does not work with the Physical Renderer if soft shadows (shadow maps) are used.