It orbits the center of mass of the galaxy, where there just happens to be a supermassive black hole. Many galaxies have central black holes, but many do not. A compact body such as this has great influence on the galactic core, as evidenced by all the frantic orbiting stars and gas in the graphic, but out where we are, the central compact object is just a tiny and insignificant part of the entire dynamical mass of the galaxy. If it magically disappeared tomorrow, we'd never notice it.

Many people seem to think that black holes are destined to gobble up everything around them, but from a distance, a black hole of say, a million solar masses, is lost in the center of mass of a 400 billion solar mass galaxy. Another way of thinking about it, if the Sun suddenly became a black hole tomorrow, it would still have one solar mass, and the planets and the rest of the solar system would maintain itheir current orbits according to Newton's Laws as if nothing had happened. Only objects very near the collapsed Sun would experience black hole-type phenomena like event horizions and infalling disks and extreme tides.

As far as the stars' orbits around the galactic center, they are roughly elliptical, but not extremely so like long-period comets. Most stars in the disk follow rosette orbits (the ends of the ellipses quite don't connect up, the ellipses don't repeat exactly, the major axes gradually precess, and the individual stars dip in and out of the galactic plane in Z as well, but for the most part they stay at roughly the same distance from the nucleus as when they were formed. The reason for this is that the stars are gravitationally coupled. Not only do they obey their combined gravity as if it were concentrated at the center, they also constantly perturb each other.

This constant self-perturbation is what forces the gas and dust to be subject to periodic pressure waves which sweep about the galaxy, alternatively expanding and attenuating the gas and condensing stellar births as the wave sweeps by. We see a snapshot of the constant propagation of these "gravitational pressure waves" which we call "spiral arms". The waves race themselves around the galaxy, but they travel at different speeds than the Keplerian motion of the revolving stellar disk population. In fact, if I'm not mistaken, the spiral does not necessarily rotate in the same direction that the stars do! It is a very different process than the inflowing feeder bands we see in, say, hurricanes, but the final result is very similar in appearance, even if it is totally different in cause.

The Galaxy is an extremely complex dynamic system, permeated with electrical, radiative, magnetic, and even acoustic fields which constantly interact with matter and with each other. It is also constantly undergoing stellar and chemical evolution, and it is criss-crossed by gradients in all these properties. We see it as a snapshot, like an instantaneous photograph of an explosion, and we are only now starting to unravel what is going on. It is as complex as a rainforest, and like a forest, it evolves on a timescale vastly slower than our own.