The subject of what galaxy lies closest to us (besides the Milky Way, which we live inside) is one of much debate, mostly revolving around what exactly constitutes a galaxy or not. Let’s go on a quick tour.
There are several nearby objects that probably constitute galaxy fragments but are no longer actual galaxies. The closest of these is Omega Centauri, 15,800 light-years away, or about 3500 times the distance to the nearest star to us.
Omega Centauri is the brightest globular star cluster in our skies—so bright that it can be resolved into individual stars with a pair of handheld binoculars, and it is easily seen as a fuzzy patch with the naked eye. Other, dimmer globulars are also visible with the naked eye, but their non-stellar nature is not easily apparent. It has about 4 million solar masses’ worth of stars, compared to the paltry half-million or so of even the massive globular clusters like M13 and M3. The only known globular with more mass is G1, sometimes called Mayall II, a globular that orbits the Andromeda Galaxy.
Omega Centauri has an abnormal distribution of metallicity, and it appears that its stars are not all the same age, unlike most globulars, which seem to form all their stars at once. This points to it being the core of a galaxy that was consumed by our own Milky Way, which would make it the closest galaxy fragment or remnant to us.
Next up is the Canis Major/Monoceros Overdensity, also called the Canis Major Dwarf, about 25,000 light-years from our own Sun. This “object” is only visible to us as a slight increase in the stellar population (particularly red giants) concentrated in an elliptical/arc-shaped area. It is completely possible that this region is a slight overconcentration of Milky Way stars and thus a mere statistical fluke. However, many scientists speculate that it is in fact a dwarf galaxy that is in the process of being shredded by our own Milky Way.
If you do not count either Omega Centauri or the Canis Major Dwarf, the closest “galaxy” to us is the Sagittarius Dwarf Spheroidal Galaxy, 70,000 light-years away. It consists of a stream or loop of material wrapped around our galaxy roughly perpendicular to the Milky Way’s spiral arms, and several globular star clusters including Messier 54, Terzan 7, Terzan 8, Arp 2, Palomar 12, Whiting 1, Segue 1, and possibly Messier 53. The Sagittarius dSph probably started being consumed by our galaxy sometime between 300 and 900 million years ago and will probably be gone entirely in the next few hundred million years. Messier 54 and Messier 53 are the closest globulars to us that do not belong to our own galaxy, and you can clearly spot them with binoculars or a small telescope; their stars are among the only extragalactic ones you can spot individually and positively identify as not belonging to our own Milky Way.
Between 6-10 dwarf elliptical and irregular galaxies rank as the next-closest galaxies to us. After that, the closest galaxy of any significant size to us (and the closest you can see the actual structure of with a telescope) is the Large Magellanic Cloud, 163,000 light-years away from us. The LMC is in fact so big and bright that it easily appears as a nebulous object under dark skies in the Southern Hemisphere. The LMC’s exact structural nature is contested; it appears to have once been a dwarf spiral galaxy and still sports a faint “bar” of stars and gas as well as telltale signs of spiral arms, but is now more like an irregular galaxy. Despite making up only 1% of the mass of our own Milky Way, the LMC is home to a number of globular clusters and nebulae, which you can observe with a backyard telescope (if you are fortunate enough to live well south of the equator).
The LMC is home to the Tarantula Nebula, a jaw-droppingly beautiful star-forming region that is the most active one known. Thanks to its extremely high activity, the Tarantula happens to also be home to SN 1987A, the closest supernova observed since the invention of the telescope, and the R136 star cluster- which contains some of the most massive and bright stars known, the biggest of which is R136A1, a Wolf-Rayet star 315 times the mass and 8.7 MILLION times the brightness of our own Sun.
The closest truly spiral galaxy to us and the closest galaxy of any significant mass is, of course, Messier 31, the Andromeda Galaxy, 2.54 million light-years away. M31 is significantly larger in physical size than our own galaxy and may or may not be more massive. M31 hosts at least 450 globular clusters orbiting around it, compared to the paltry 150–180 or so our Milky Way has. M31 was the first galaxy proven to exist outside our own and the first to have its distance measured, and it can be seen with the naked eye or binoculars even from suburban areas. Binoculars or a small telescope will reveal M31’s companion galaxies, M32 and M110 (both are dwarf elliptical galaxies), as well as a hint of the galaxy’s dust lanes. A large telescope can show you the immense globular cluster G1, located several degrees away from M31 in the sky, as well as M31’s brightest open star cluster NGC 206 and maybe a bit of the galaxy’s spiral arms.