Apostrophes serve three functions: they form possessives, they indicate contractions and the omission of letters or numbers, and they can indicate plurals of single letters.
To indicate the possessive case for singular nouns, add ’s: Burt’s dog, Charles’s apple, Cleopatra’s realm. Although The Elements of Style suggests appending only an apostrophe to form the possessive of biblical and ancient names ending in -s (e.g., Jesus’, Socrates’), this exception is only loosely defined and has likely given rise to the misconception that singular nouns ending in -s form the possessive with only an apostrophe. Prefer the consistent, modern approach, which The Chicago Manual of Style encourages (thus Jesus’s, Socrates’s), unless your style guide dictates otherwise.
Plural words ending in -s form the possessive with only an apostrophe; all other plural words take ’s: guys’ night out, the geese’s flight. The same applies to singular mass nouns ending in a plural -s: Five Guys’ location, United States’ role.
Note that, according to the U.S. Board on Geographic names, most American place names do not use the possessive apostrophe on the contentious basis that names such as Pikes Peak and Toms River should indicate a “single denotative unit” and not a possessive location. This rule has five exceptions: Martha’s Vineyard, Ike’s Point, John E’s Pond, Clark’s Mountain, and Carlos Elmer’s Joshua View.
Except for lowercase letters, do not form plurals with an apostrophe: The 1990s were a riot, We celebrate many Christmases, I have many e’s in my name. Plurals of uppercase letters do not require an apostrophe but using one can avoid confusion: I have two As and a B, A’s are not the most common letter in the alphabet (the latter can create a miscue without an apostrophe).
Contractions and letter omissions require an apostrophe: don’t, ne’er, shouldn’t’ve. Beware that certain contractions lead to easily-mistaken homonyms: its and it’s, whose and who’s, their and they’re.