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
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.
If you want to know the number of
But the actual number of galaxies that have been observed is order of magnitudes larger.
The case I know is the Sloan Digital Sky Survey 3 (SDSS3):
they observed about a third of the sky using five different filters between optical and near infrared. Of all the objects detected
To these number you have to add the galaxies observed by surveys in different areas of the sky, deeper that SDSS (which means that they can seen fainter and/or more fare away objects).
By the way, if memory does not fail me, estimates gives about 10^10 galaxies in the observed universe