There exist many astronomical catalogs, reaching from rather simple catalogs like the Bayer catalog for stars, or the Messier catalog for galaxies, nebulae, etc., to huge catalogs with millions of entries. A catalog with about a billion objects is scheduled for 2022.
There exist also printed editions of some catalogs.
Objects in the sky have several properties, by which they can be described and mostly identified.
One of the most frequently used properties is the position in the sky. From the local geographic position and the time, the observed position of the object can be translated into a celestial coordinate system, e.g. the equatorial coordinate system. The position of an object is described in this system by declination and right ascension.
Two other frequently used properties are magnitude and color.
Also frequently used are patterns of stars, meaning their relative position to each other, beginning with star constellations.
The analysis of steller spectra allows a more detailed identification.
Longer term observations may uncover variability and proper motion as additional properties.
Here is a description of parameters (including properties) provided by the BSC5p catalog.