A galaxy is, by definition, any large collection of stars that can be recognized as a distinct physical entity. In terms of the number of stars, a small 'dwarf irregular' galaxy like the Small Magellanic Cloud, has about one billion stars in it, but there are even smaller systems that are recognized as galaxies such as the Leo I and II dwarf galaxies with about 1 million stars in them, and the Draco System with a few hundred thousand stars in it. The largest star cluster, a globular cluster called Messier 15 has about 6 million stars, so we see that for small galaxies, there is a blurring together of what we mean by a galaxy and a large star cluster.
In addition to their mass and numbers of stars, a galaxy is a collection of stars and gas which move through the universe independently of the Milky Way. Globular clusters are roundish swarms of stars that orbit the Milky Way, while the Leo and Draco Systems seem to be independent collections of stars.
NGC 4414, a typical spiral galaxy
Many galaxies also continue to form new generations of stars. The Milky Way, and all spiral shaped galaxies like it (see above image of NGC 2997), produce new stars at a rate of one or two stars per year. These stars are formed in the vast interstellar clouds that account for about 1 to 10 percent of the mass of these galaxies. Globular star clusters, on the other hand, are not currently forming stars because this activity happened billions of years ago and then stopped once all of the gas and dust clouds were used up.
Source: wikipedia, astronomycafe.net