Many stars and other objects beyond the Solar System have proper names, but the vast majority of listed objects have only designations in one or more catalogs, and the vast majory of all the billions or trillions of objects observable with modern instruments have not been listed in any catalog and have no designation.
So how many catalogs of astronomical objects are there?
J.R.R. Tolkien wrote in the first chapter of The Hobbit:
Gandalf! If you had heard only a quarter of what I have heard abou thim, and I have heard only a very little of all there is to hear, you would be prepared for any sort of remarkable tale.
And if somone knew of only a quarter of the astronomical catalogs that I know of, and I know of only very few of the total number, they would know there is a large number of such catalogs. I will only mention a few.
Part One of Two: Star Names and designations.
There have been many thousands of cultures speaking thousands of languges on Earth. And most cultures had proper names for at least a few of the 6,000 to 8,000 stars which can be viewed by the naked eye from Earth.
I own a copy of a book Star Names Their Lore and Meaning by Richard Hinckley Allen, 1899, 1963. And you might think that it would be nice to have a little booklet listing the names of stars and constellations in various cultures and maybe ten or twenty pages long. Star Names Their Lore and Meaning has XIV pages of titles and introduction and 563 numbered pages.
There is a book The Stars of Jade by Julius D.W. Staal, about Chinese constellation and star names.
And no doubt there are many other sources for proper names of stars.
In 1603 johann Bayer published Uranometria, an atlas of the constellations. Most of the stars were marked with Bayer designations, consisting of a lower case Greek letter and the genitive case of the constellation name in Latin.
For the benefit of people unfamiliar with the Greek Alphabet, Bayer designations usually have the Greek letter spelled out: Alpha, Beta, Gamma, Delta, and so on down to Omega, when there are that many visible stars in the constellation. If there were more stars than Greek letters, Bayer used an uppercase "A" followed by lower case b, c, d, etc.
Bayer mostly lettered the stars in order of brightness as seen from Earth, but sometimes by position in the constellation.
Most stars visible from the northern hemisphere have Flamsteed designations from the 1712 and 1783 editions of Historia Coelestis Britannica.
Flamsteed designations for stars are similar to Bayer designations, except that they use numbers instead of Greek and Roman letters. Each star is assigned a number and the Latin genitive of the constellation it lies in (see 88 modern constellations for a list of constellations and the genitive forms of their names). Flamsteed designations were assigned to 2554 stars. The numbers were originally assigned in order of increasing right ascension within each constellation, but due to the effects of precession they are now slightly out of order in some places.
So many of the stars visible from the northern hemsphere now have proper names in various languages, Bayer designations, and Flamsteed designations.
And since then tens and hundreds of different star catalogs have been made. Stars in those catalogs are usually identified by the name or abbreviation of the catalog followed by the number of the star in the catalog. Of course many stars have designations designations in several different catalogs.
As a result the nearest stars to the Sun include:
Alpha Centauri & Proxima Centauri, Barnard's Star, Wolf 359, Lalande 21185, Luyten 726-8, Ross 154, Epsilon Eridani, Lacaille 9352, EZ Aquarii, Struve 2395, Groombridge 34, and so on.
Since 2016 the Working Group on Star Names of the Internatinal Astronomical Union has been approving official proper names for stars and has approved 336 proper names of stars by August 2018.
Part Two: Names and Designations of Deep Sky Objects.
Astronomers noted hazy patches of light in the sky.
French Astronomer Charles Messier was interesting in finding comets, so he made a list of hazy objects in the sky so he wouldn't mistake them fom comets, and published his list from 1774 to 1781. The various Messier objects he cataloged are designated as M1 through M110.
Other astronomers published catalogs of such cloudy or nebulous objects, including John Dreyer's New General Catalog of Nebulae and Clusters of Stars, 1888, followed by two Index Catalogs in in 1895 and 1908.
Most of the well known deep sky objects are in Dreyer's catalogs and are listed as NGC or IC followed by their numbers in the catalog.
As it turned out, the deep sky objects listed in those catalogs belong to several vastly differen types.
One type is diffuse nebulae, which are "clouds" of gas and dust in interstellar space, often many light years across. They include emission nebulae which are heated by stars and so emit light, reflection nebulae which reflect light from stars, and dark nebulae which block light from stars and nebulae beyond them.
Another type is planetary nebulae, which are shells of gas emitted from stars collapsing into white dwarf stars at the end of their evolution.
Another type is open clusters, clusters of young newly formed stars within star forming regions of a galaxy.
Another type is globular clusters, much larger clusters of usually older stars, that obit galaxies outside of the main bodies of those galaxies.
Another type is galaxies. Galaxies are vast associations of stars, star clusters, and dark matter, Many galaxies also have a lot of gas and dust.
Galaxies range in size from tiny dwarf galaxies which don't have many more stars than globular star clusters to supbergiant galaxies with millions of times as many stars.
The main types of galaxies are spiral, elliptical, and irregular.
And now that many types of deep sky objects are known, there are many catalogs listing only a single type of deep sky object.
So emission nebulae, reflection nebulae, planetary nebulae, supernova remnants, planetary nebulae, open clusters, globular clusters, stellar associations, dwarf galaxies, giant galaxies, spiral galaxies, elliptical galaxies, irregular galaxies, clusters of galaxies, superclusters of galaxies, cosmic radio sources, cosmic X ray sources, neutron stars, pulsars, black holes, quasars, et. etc. all have separate catalogs.
And any object listed in such a catalog should have its catalog designation, and many objects are listed in more than one catalog and have more than one catalog designation.