Excellent question, and your right, our galaxy is the Milky Way! Your question can be answered with one word: gravity, but I like to ramble about science, so I will go on. All things in the universe that have mass are attracted to each other by the force called gravity. Early in the life of our universe, shortly after the big bang, particles of matter started to attract other particles. Eventually these balls matter (atoms) form huge bodies like stars. Stars attract other stars because of gravity and eventually form galaxies, which are just huge groups of stars. Galaxies get bigger and more massive when they capture stars. This makes the gravity of the galaxy stronger and it captures more and more stars. This is how it grows. Most galaxies have black holes at their centers. Black holes are balls of matter that are so dense and have such strong gravity that not even light can escape them; that is why they are dark. Galaxies get there disk shape because they are rotating. The groups of stars flatten into a disk as the galaxy spins. This is the same reason that our solar system is flat. The flattened disk of spinning matter that solar systems form from is called a nebula.

You should check out the NASA website on galaxies gal axies It has some neat videos that show how galaxies form.