Is it James's or James' — how do apostrophes work for possession?

Both are defensible, and the choice depends on which style guide you follow. The wider rules for possessive apostrophes are more settled.

Singular nouns add apostrophe + s: the dog's bowl, the company's policy, a day's work.

Plural nouns ending in s add only an apostrophe: the dogs' bowls, the companies' policies, two days' work.

Irregular plurals not ending in s add apostrophe + s, because they behave like singulars: the children's toys, the women's team, the people's choice.

The contested case is a singular name already ending in s — James, Charles, Dickens. Two conventions exist. Most modern guides, including Chicago and Oxford, prefer James's, on the grounds that you pronounce the extra syllable so you should write it. Journalism style guides, including AP, often prefer James'. Both are widely used; neither is an error.

A useful test for the ambiguous cases: say it aloud. If you pronounce an extra iz sound, most guides say write the s.

Two rules that are not negotiable:

Possessive pronouns never take an apostrophe — its, hers, ours, yours, theirs, whose.

Plurals never take an apostrophe. Apple's £2 on a market sign is wrong; so is the 1990's. Write apples and the 1990s.

For joint possession, mark only the last name: Ravi and Priya's house means one shared house. Ravi's and Priya's houses means two.

Pick a convention for names ending in s, and hold to it throughout a document.

One further case: for a possessive of a phrase or a compound, mark only the end. My brother-in-law's car, the Secretary of State's decision. Trying to mark the head noun instead produces the awkward my brother's-in-law car, which nobody writes.