The Great Debate, more romantically (there was a prior writeup about something from The David Letterman Show being called "The Great Debate". That writeup is gone, thus I strike my somewhat bitchy, but pithy aside), refers to an astronomical symposium held in April, 1920 at The Smithsonian's Museum of Natural History. A debate entitled The Scale of the Universe propelled mankind toward a deeper understanding of cosmology and our place in the universe.
Heber D. Curtis theorized that the universe consisted of many galaxies like our own Milky Way. Harlow Shapley disagreed, theorizing that our galaxy was the only component of the universe.
At the center of the debate were astronomical phenomena known at the time as spiral nebulae. Curtis argued that these objects were extragalactic and were in fact galaxies unto themselves. Shapley stated his belief that these "nebulae" were just nearby clouds of interstellar gas.
Shapley's model of the "Great Universe" (or "Great Galaxy") placed our sun far from the center of the galaxy, whereas Curtis' model placed the sun near the center of a far smaller galaxy.
The debate was at least partially resolved in the mid-20s by Edwin P. Hubble, who, by using the Hooker Telescope at Mount Wilson Observatory (the largest telescope in the world at the time), proved that the distance from Earth to M31 (also (now) known as the Andromeda Galaxy) was far greater than even the size of Shapley's theoretical universe. Hubble's study of Cepheid variable stars proved that M31 was, indeed, outside our own galaxy.
By the 1930s, increased understanding of globular clusters and interstellar absorption indicated that the size of the galaxy, and the very universe, had been very seriously underestimated. At that time, it was accepted that Shapley was correct in his theory that the sun was toward the outer edge of the Milky Way, and Curtis was correct in his theory that the universe contained multiple galaxies of which the Milky Way was merely an insignificant part.
In 1996, another debate entitled "The Scale of the Universe" was held, and at the same place as the original. In this debate, Hubble's Constant, or the rate at which the universe is presumed to be expanding, was argued.
The Great Debate has gone down in history as one of the defining moments of modern astronomy, and a prime example of how theoretical astronomy is truly "way out there".