How many galaxies have been discovered? I know Andromeda is our nearest galaxy. But how many known galaxies have been discovered and what are they?
First, the nearest galaxy to ours is not Andromeda, according to the NASA based page "The Nearest Galaxies", it was until recently considered to be the Canis Major Dwarf Galaxy at approximately 42,000 light years away.  However, recently, there is suggestion that the closer Omega Centauri Globular Cluster may be a disrupted dwarf galaxy (Another source).

Now, in answer to your other query of how many galaxies there are - there is a global project called Galaxy Zoo which is attempting to catalogue and classify all observed galaxies, so far, according to the Galaxy Zoo for astronomers page, thus far they have


  
  the entire Sloan Digital Sky Survey spectroscopic sample and all existing Hubble Space Telescope surveys (around 1.5 million galaxies in total).
  


Pictured: The Hubble Ultra-Deep Field which contains over 10,000 objects, the majority of which are galaxies. The image is only a bit over 3 arcminutes across--a tiny sliver of the sky.