Writers use the apostrophe to substitute for the preposition of, by, with, or for:
a doctor’s appointment (appointment with the doctor)
a week’s notice (notice of one week)
the children’s toys (toys for the children)
Presidents’ Day (Day for [two] Presidents)
the boss’s desk (desk of the boss)
Sophocles’ plays (plays by Sophocles)
everybody’s concern (the concern of everybody)

Use the apostrophe and -s in these instances:
For singular nouns before plural nouns or other singular nouns: attorney’s fees, the book’s editor
For singular nouns ending in –s: Bill Gates’s computer OR Bill Gates’ computer (unless the pronunciation is distorted: Ulysses’ computer, NOT Ulysses’s computer)
For indefinite pronouns: nobody’s business
For joint ownership: Ted and Jane’s wedding
For individual ownership: Ted’s and Jane’s wedding rings
For compound singular nouns: the lieutenant governor’s staff
For compound plural nouns: my brothers-in-law’s jobs
For words with fixed apostrophes: bachelor’s degree, bachelor’s degrees
For plural of letters, numbers and symbols: 3’s and A’s.
For words used as words: too many wherefore’s in legalese; no if’s, and’s, or maybe’s
For time periods: 1900’s OR 1900s
For plurals of initials and abbreviations: two Ph.D’s on the faculty; several TV’s or TVs

Use only the apostrophe in these instances:
For plural nouns ending in –s: the Joneses’ van, my parents’ divorce, both students’ papers, all instructors’ grades
For plural nouns not ending in –s: children’s toys, the people’s court, women’s rights

Use an apostrophe for contractions:
it’s (it is), who’s (who is), you’re (you are), they’re (they are)

Use a possessive noun or pronoun preceding a gerund:
my going to college, your coming to class, Carol’s leaving early, Mandy’s being on time, your giving a report, their being here

Do not use an apostrophe
For possessive pronouns: its, whose, yours, theirs, ours, hers
For plurals: the Joneses, dictionaries, bonuses
For nouns used as adjectivals: Tina Turner legs, Betty Davis eyes, a Roy Jones fight, student lockers, employee benefits, Choir Day, an Emily Dickinson poem, a Steven King novel