One of the things I love about astronomy is you don’t actually need to know anything to appreciate the night sky. But if you want to communicate with others about what you saw or learn what’s out there, you have to have names.
Ever since humans have seen the stars, they’ve seen patterns. These patterns are called asterisms and they can be whatever you want. Some are common and have established names, such as the Big Dipper and Orion’s Belt. It’s important to remember that even though stars appear close to each other in the sky, they’re (usually) not actually connected. Since the stars are so distant, they all look equally far away, but they’re not. It would be like standing in New York City and thinking that because Chicago and San Francisco both lie in the same general direction, that they were closely connected.
Ptolemy formalized 48 of these as constellations. The International Astronomical Union (the international authority for astronomy) modified and expanded this list (including adding constellations that can only be seen from the southern hemisphere) into the modern official 88 constellations. Furthermore, constellations are now defined as areas of the sky, not as patterns of stars. This means that every point in the sky is part of some constellation. The bright stars may form patterns that help us to identify it, but they are not the constellation per se. I like to think of the stars as cities, and the constellations as states or countries. In the United States, some states have several large cities, others may have none. But every point in the United States is part of some state.
The names of stars are bit more complex, since there are an estimated 3 sextillion stars in the observable universe (3 × 1023, or almost a trillion trillion stars)! Clearly, we need some naming scheme. I’ll normally use Johann Bayer’s method. To each star in a constellation, he assigned a lower-case letter from the Greek alphabet, usually following brightness and location characteristics. (After he ran out of Greek letters, he switched to lower-case Latin letters — our alphabet — and then upper-case letters). In his designation, the name of the star consists of the letter followed by the constellation name in the “genitive case” (a possessive form). Either the Greek letter can be used or it can be spelled out as an English word, and either the full constellation name or the standard three-letter abbreviation can be used. So for example, take Rigel, in the constellation Orion. This star will be called Beta Orionis, or β Orionis, or β Ori for short. Many stars also have their own names (like Rigel!). For the twenty or so brightest stars, their individual names are commonly used, but for the rest, Bayer designations are most common. There are stars that Bayer didn’t list, and for those I’ll use John Flamsteed’s names: they’re similar to Bayer’s method, but use a number instead of a Greek letter (for example, 40 Eridani, one of the stars in Eridanus).
For a more thorough discussion, see Sky & Telescope’s excellent Names of the Stars.