There are about 50 galaxies in the Local Group (see list of nearest galaxies for a complete list), on the order of 100,000 in our Local Supercluster and an estimated number of about 170 billion in all of the observable universe.
The discovery of the nature of galaxies as distinct from other nebulae (interstellar clouds) was made in the 1920s. The first attempts at systematic catalogues of galaxies were made in the 1960s the Catalogue of Galaxies and Clusters of Galaxies listing 29,418 galaxies and galaxy clusters, and with the Morphological Catalogue of Galaxies, a putatively complete list of galaxies with photographic magnitude above 15 comprising 30,642 items.
In the 1980s, the Lyons Groups of Galaxies listed 485 galaxy groups with 3,933 member galaxies. Galaxy Zoo is a project aiming at a more comprehensive list: launched in July 2007, it has classified over one million galaxy images from The Sloan Digital Sky Survey, The Hubble Space Telescope and the Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey.
There is no universal naming convention for galaxies, as they are mostly catalogued before it is established whether the object is or isn't a galaxy. Mostly they are identified by their celestial coordinates together with the name of the observing project (HUDF, SDSS, 3C, CFHQS, NGC/IC, etc.)
NOTE: Above information has been taken from wikipedia and/or official websites of topics.