In general, which is used to introduce nonrestrictive phrases. If you take out a nonrestrictive phrase, the sentence should still make sense and have the same meaning.

If you take out which haunts my nightmares, the sentences still retains its meaning: This ugly leopard-print chair is my husband’s.

That is used to introduce restrictive phrases. A restrictive phrase is essential to the meaning of the sentence.

If you take out the phrase that looks as though a leopard died on top of it, then the resulting stub (The chair is my husband’s) no longer identifies the chair properly. The phrase that looks as though a leopard died on top of it restricts the field of all possible chairs to identify the particular chair the speaker has in mind.

If you’re struggling to decide whether which or that is correct, remember this basic rule: if you see commas, use which. If you don’t, use that.

When discussing people rather than things, use who or whom instead of which and that. See Who vs. Whom for more on the difference.

It’s okay to use that to refer to people, but which only works for things.