Matcap stands for “material capture” and is an image that is used as an image texture to fake a whole material including lighting and reflections in 3D applications. The most common use is while sculpting objects but it can be used in games for objects that should always be lit or to achieve special effects such as x-ray, ghost or radar.
The image consist of a simple sphere that is mapped to the object’s normals in relation to the camera. As the camera moves around the object, the reflections and highlights move around your object much like the object where moving and not the camera. In other words, if your object were a sphere, no matter how you looked at it, it would look like the matcap sphere (shading, colors , reflections etc. would always be displayed in the same place). But when a matcap is applied to non-spherical shapes the material responds as if it were made of a complex material.
Matcaps is nice while sculpting. It do work with rendering, to an extent, as the material do not interact with the scenes lightning and reflection of other objects in the scene is best used for proof of concepts. It can be, and is, used to achieve special effects in games. It’s a good time saver when you do not have the time to set up complex lighting or materials.
The best way to figure out matcaps is to fiddle with them, so go on and play!