Lie and lay would be easy if it weren't for one cruel overlap: the past tense of lie is spelled lay. Everything else follows from a simple split.

Lie is something you do by yourself. People lie on beaches, books lie on tables, the answer lies in the data. There's no object; nothing is being placed. Its forms are irregular: lie, lay, lain, lying. Lay is something you do to something else. You lay a table, lay bricks, lay a hand on someone's shoulder. It always has an object, and its forms are regular apart from the spelling: lay, laid, laid, laying (never "layed").

That's how the trap springs. "Yesterday I lay down" is perfect English, because lay there is the past of lie. "Yesterday I laid down" is the mistake, unless you laid something down. And the rare past participle lain ("it had lain untouched for years") sounds so formal that people reach for laid instead.

There's a third verb hiding here too: lie meaning to say something untrue. It's regular and causes no trouble: lie, lied, lied, lying.

In relaxed speech, "I'm going to lay down" is everywhere, but learner's dictionaries still mark it as incorrect. In writing that will be judged, keep the two apart.

Memory trick. Lay has an a, like place: you lay something in place. Lie has an i, like recline: you lie when you recline.