Billions of galaxies remain in the known Universe, and not any two are exactly the same. Galaxies are large compared to what we can comprehend, some thousands of light years across (a light year is approx 300,000kms a second!), others are hundreds of thousands of light years from one side to the other.
Each galaxy is made up of tens of millions of stars, sometimes trillions of stars, and holds enough gas and dust to build another galaxy.
Most galaxy counts are done with the aid of pc's these days; we no longer employ folk to sit up all night counting them by hand!
Star counts for galaxies can be calculated by a a good number of ways, one of which is by dividing the total luminosity of a galaxy, by an average stars mass.
The distance between galaxies is usually millions of light years, but we can see collisions between them, and interactions as they pass each other on their endless journey in various parts of the Universe. Our own galaxy, the Milky way, will at some point collide with Andromeda, are largest partner.
While there are a number of different types of galaxies, they can be classified into four areas;
Spiral (examples Andromeda, Milky way):
These are flattened disks with a spiral pattern within the disk. They have a large bulge in the center, with what is believed to be a black hole in the centre, and from this bulge come the spiral arms which reach out to the edge of the galaxy. It can usually be clearly seen in pictures that around the outside are huge gas and dust clouds.
Their rotation is quite ordinary compared to a lot of things in the galaxy, always rotating in the same direction around the fundamental bulge. Generally you find the older stars in the center, while new young blue stars are forming amongst the gas and dust around the edges.
Barred (examples M95, M91):
This is similar to the Spiral, apart from the arms are more straight reaching from the bulge to border of the galaxy, rather than curved. Various astronomers now believe that the Milky way we live in is a barred galaxy.
Some of these can look irregular, but closer inspection, such as the Magellanic Cloud show bars within it, distorted by interactions with other galaxies in the local cluster we are in.
Elliptical (examples M60, M105):
These are generally smooth in appearance, and unlike the two galaxy kinds above, the stars in them can be moving in any direction, and they have very little gas and dust within them, so this means very little star formation. Something that will result in a quicker death than spirals, while 'quicker' as in billions of years!
Irregular (examples Leo a,Wolf-lundmark-melotte):
These have no structure at all, and you can have millions of stars in clusters, then a space, another clump, etc. Most have little star formation; others are full of gas and dust and have very considerable star birth rates.
Out of all the galaxy kinds, these are usually the hardest for the amateur astronomer to pick up due to the overall lack of brightness. Also these galaxies are usually nowhere near as big as the other kinds, with most being called dwarf.
It is more than likely there are a lot more of these galaxies than the others, but until full computerised sky surveys have taken place we do not know at the moment, and we may never be able to see them all due to the lack of luminosity from them.