'Lay' has an object; it always acts on something; you lay something down. It could be your keys on the counter, a bag on the ground, or yourself on a bed. It's also the past tense of 'lie', which probably confuses a lot of people. So, for example, you could say "I lay myself down in bed" or "I lay there all last night." The past tense of 'lay' is 'laid'.
'Lie' has no object; it's reflexive. After you lay an object down, if it's inanimate it will lie there until someone picks it up. i.e. "You look tired; you should lie down awhile."
Hopefully this clears up some confusion. Now go and lay down the law...s of grammer.