lay is a transitive verb; it takes an object. For example,
Lay the books on the sofa.
Present: Chickens lay eggs.
Past: Chickens laid eggs.
Participle: Chickens had laid eggs . . . .
lie is an intransitive verb; it doesn't take an object. For example,
Lie on the sofa.
Present: Chickens lie in the sun.
Past: Chickens lay in the sun.
Participle: Chickens had lain in the sun . . . .
lay is the past tense of lie. Some speakers, if not more, though, do in fact use laid instead of lay; e.g., "He laid down last night at about 9:00", but it's not considered Standard English or what those who subscribe to traditional grammar rules would consider acceptable English. Nonetheless, that speakers - and I am one - use transitive laid down to mean intransitive lay down is a matter of (a) hyper-correction: as a past tense verb, lay just doesn't sound all that correct, so speakers add -ed to lay to give it that ol' past tense feeling, and (b) semantics: for some speakers, to lay (oneself) down is transitive because it's reflexive; e.g., "He laid (himself) down last night at about 9:00."
In short, lay, not laid or lied, is the past tense of lie.
Past: lay (note, but speakers are using laid.)
Read more here: http://www.ku.edu/~edit/lie.html