A Material is an asset that can be applied to a mesh to control the visual look of the scene. At a high level, it is probably easiest to think of a Material as the "paint" that is applied to an object. But even that can be a little misleading, since a Material literally defines the type of surface from which your object appears to be made. You can define its color, how shiny it is, whether you can see through the object, and much more.
In more technical terms, when light from the scene hits the surface, a Material is used to calculate how that light interacts with that surface. These calculations are done using incoming data that is input to the Material from a variety of images (textures) and math expressions, as well as from various property settings inherent to the Material itself.
Unreal Engine 4 utilizes a physically-based shading model. This means that rather than defining a Material using arbitrary properties (such as Diffuse Color and Specular Power), you instead use properties more easily relatable to the real world. These include Base Color, Metallic, Specular, and Roughness.