The word "hyperbolic" in the title refers to the coarse geometric property of having uniformly thin geodesic triangles. A group is hyperbolic if it acts properly cocompactly on some hyperbolic space (eg. the hyperbolic plane). A subgroup $H\leq G$ is hyperbolically embedded if there is a "relative" Cayley graph for $G$ which is hyperbolic, in which $H$ satisfies some weak local finiteness property. I'll explain this in more detail and give some examples and applications.