The bright parts around the black hole are the accretion disk, which is in reality just a flat disk in the equatorial plane similar to the rings of Saturn, but is distorted visually by gravitational lensing. You can see a page here that gives some code for creating images using ray-tracing of light rays in curved spacetime, which offers a more schematic diagram of the visual appearance of a disk around a black hole (with a checkerboard pattern on it for clarity):
In this Q&A with Kip Thorne, he gives some background on how they created the images, indicating that they used a more sophisticated technique than ray-tracing:
I had been seen many years ago an image of an accretion disk with gravitational lensing that Jean-Pierre Luminet in France had made. I had sort of forgotten about it, but when I first saw the gravitationally lensed accretion disk that you actually see in the movie, it was a mixture of amazement on one hand and recognition that “Yes I do remember seeing something like that, years ago.” And a bit of awe and excitement that this team at Double Negative had just taken the equations I had given them — they don’t just use ray tracing, they propagate ray bundles or light beams — they’d used light beam propagation equations, laid down their own accretion disk based on artistic models based on astrophysicist’s stuff, and come back to me with a full-blown image of the sort you see in the movie. I was really impressed and gratified that they pulled it off and was so pleased with how it looked.
They didn't simulate all the optical effects that would be seen though--the physicist mentioned above, Jean-Pierre Luminet, comments in a facebook post here that the Interstellar image doesn't include "the strong Doppler and gravitational spectral shifts induced by the rotation of the disk at relativistic speed", and that after commenting about this he got a message from Kip Thorne saying that "The doppler shift was left out of the images, because (as you showed long ago) it makes the disk highly asymmetric, and much harder for a mass audience to grasp." Luminet also links to his original 1979 paper on the visual appearance of a black hole with a thin accretion disk, and also gives this image "computed by J.A.Marck in the 1990's" which does take into account the Doppler shift, as you can see it's a bit less cinematic:
I also found this video linked to and explained in this reddit post, showing a schematic image of what it would look like to fly by a black hole with an accretion disk at relativistic speeds. The color scheme is artificial, different colors represent different light intensities.