A galaxy is what you get when gravity gathers stars — anything from a few thousand to a few hundred billion or more — into one turning system and keeps them there. There are hundreds of billions of them. On a dark autumn night you can find one of them with no equipment at all.
Stars are not scattered evenly through the universe. They come in enormous clumps, held together by gravity, with almost entirely empty space between the clumps. One clump is a galaxy. Ours is called the Milky Way, and it is an ordinary one — big, but nothing special.
They come in a handful of shapes, and the shape tells you the history. Spirals are still making new stars in their arms. Ellipticals have run out of the gas to make new ones, and are coasting on the old ones. Irregulars have often been pulled out of shape by a neighbour — and some have been through an actual collision.
Andromeda is about the far end of your own eyesight: on a moonless autumn night, with a clear view to the east (from the northern hemisphere), it is a faint smudge to the naked eye. That smudge is about 2.5 million light-years away, which means the light entering your eye left before there were any people like us.
Expect a smudge. Almost every galaxy photo you have seen took hours of exposure, usually on a telescope far bigger than yours. Through a small one, a galaxy is a pale grey oval with no colour and, usually, no visible arms. Knowing that in advance is the difference between a good first look and a disappointing one.
Far enough that the question was genuinely open until about a hundred years ago — people argued about whether these smudges were inside our own galaxy or were galaxies themselves. The answer came from a particular kind of star that pulses at a rate which gives away how bright it really is.