Is it any wonder so many people don’t know whether to say “lie” or “lay” when they see stuff like this? It is said wrong or spelled wrong so often the mistake almost becomes the rule.
The technical explanation is that “lie” is intransitive and “lay” is transitive, i.e. it takes an object. So you always lie down. You don’t lay down, not unless you have a bag of duck down and lay it on the bed. The confusion comes from the fact that the past tense of lie is lay. “Yesterday I lay down” is correct. “I’m going to go lay down” is not. “Lay down your gun,” is correct because gun is the object.
What about “Now I lay me down to sleep”? That’s correct, too, because of the “me” in the sentence. That’s the object. Without it the sentence would need “lie”, but that would spoil the meter, which why the author chose “lay me” instead. For the participle of lie, use lain. “I have lain here for over an hour but can’t fall asleep.”
That only leaves “lay” to deal with. Simple: lay, laid, laid. “Lay an egg,” “She laid an egg,” “She has laid an egg.” Simply put, the Adam@Home comic above laid an egg.