The Andromeda Galaxy is one of the galaxies that is closest to the Milky Way Galaxy, lying some 1.5 million lightyears away. (Depending on how you count, it is actually the closest, since its two competitors, the Magellanic Clouds, are satellite galaxies of our own, rather than full galaxies). It is a spiral galaxy, much like our own galaxy, and is on about the same scale as our galaxy. It is also moving towards our galaxy, and in a few billion years will collide with us.
The Andromeda Galaxy is also the furthest object visible to the naked eye (if you have good eyesight and are free of light pollution). It is found in the constellation Andromeda, which is fairly far north, although not far north enough to be circumpolar for most of North America or Europe. The easiest way to find the Andromeda Galaxy is to find the prominent "W" of Cassiopeia, which is a circumpolar constellation. Following the arrow of the second "V" of the "W" down points towards the constellation Andromeda, and also to the Andromeda galaxy. For your efforts you are rewarded with...a blurry white patch. Whether viewing this blurry patch gives you a feeling of cosmic distance is probably more up to your psychology at the moment than anything intrinsically interesting about it. With binoculars or even a small telescope you can see more detail, and may even begin to see some of the spiral structure. With an expensive telescope, those details quickly become apparent.
Since viewing it with the naked eye, or even with early telescopes, didn't show the Andromeda Galaxy that early, most early astronomers just thought it was another nebula. It was Immanuel Kant, in the late 1700s, who first proposed that it was a galaxy much like the Milky Way. He termed it an Island Universe. When better telescopes were developed, his guess was confirmed to be correct.