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A photon sphere is a sphere-shaped (or more often, an oblate-spheroid-shaped) surface-region around a black hole indicating the distance at which photons can orbit the black hole, i.e., any orbiting photon will be located on it. For a non-rotating black hole, it is 3/2 the Schwarzschild radius from the center of the black hole. Within the photon sphere, no object can maintain an orbit without accelerating (e.g., assisted by a force): it will either fall into the event horizon or escape the black hole. This phenomena is entirely due to the strong-field gravity, and in theory (i.e., if its equation of state would support such a density), a neutron star could be dense enough to have an external photon sphere: I assume that if it were non-rotating, it would also be at 3/2 its mass's calculated Schwarzschild radius. For a rotating black hole, there are two nested photon spheres, one within the ergosphere and another external to it.
Photons that happen to be quite close to the altitude of the photon sphere may encircle the black hole one or more times before escaping, which affects the black hole's appearance: illumination coming from a ring-shape that surrounds the black hole.