Ask a room of professional editors to name the hardest pair in ordinary English and most of them will say this one. Not the rule — the rule is manageable. The past tense. Because the past tense of "lie" is "lay". The same word as the present tense of the other verb. So somebody who has learned the rule correctly, and applies it correctly, writes "she lay down" and then stares at it wondering whether they have just made the mistake they were avoiding. They have not. But the correct answer looks exactly like the error, which is why this pair defeats people who have no trouble with any of the others.
A transitive verb — it needs an object. To put something down. Past tense: laid.
An intransitive verb — no object. To recline, or to be situated. Past tense: lay. Past participle: lain.
Quick way to remember
Ask whether there is a direct object — something being put down. If yes, use "lay": you lay a book on a table. If nothing is being placed, use "lie": you lie on a sofa. For the past tense, remember that yesterday you lay down, but you laid the book down.