The term f(R) gravity refers to a particular type of theory of gravity that is a modification of general relativity (GR), the current well-accepted model. They encompass some of the modifications to GR being explored as means of explaining current cosmology data (particularly the data motivating dark energy) and as a step toward combining GR with quantum mechanics for a Theory of Everything. The gist of f(R) gravity and the motivation for its name is that in some equations, it plugs in a function of R (R being the Ricci scalar) in an equation term where "normal" GR has just R itself. Some consequences can be worked out based on merely this fact, that the term replacing the Ricci scalar is dependent on the Ricci scalar but independent of other factors, which gives some opportunity for exploration of such theories. By this definition of f(R) gravity, some otherwise-explored alternatives can be classified as f(R) gravity theories, but generally go by their already-established names. Another class of recent gravity theories is termed scalar-tensor gravity (and I don't know whether these two classes of theory are mutually exclusive).
An additional motivation for the development of such modified GR theories is that as our ability to observe black holes continues to grow (such as the development of the Event Horizon Telescope, gravitational-wave detectors, the refinement of techniques to detect black hole characteristics through observation of galaxies and the constant development of telescopes and instruments), opportunities grow to eliminate or confirm these theories.