You're talking about three separate verbs here. The first is lie (past tense: lied, past participle: lied). It means ‘not tell the truth.’ I can’t stand it when she lies to me The second is lie (past tense: lay, past participle: lain). It means ‘lie down.’ I wanted to lie down when I got home from work. I lay in my bed all day. He has lain on the couch for a little over an hour. Lie can also mean ‘originate from (something),’ as in: The problem lies in the social injustices of this country. The third is lay (past tense: laid, past participle: laid). It means ‘put (something) down.’ He laid the box on the table began to open it. It can also mean ‘prepare (something) in a position,’ as in: lay a fire, or lay the foundation for a new enterprise Another meaning of lay is ‘impose (something) on someone’ He laid the responsibility on his peers. Personally, I hardly ever use ‘lay’ in the senses listed above when I am speaking. ‘Lay an egg’ is a common expression used when talking about oviparous (egg-bearing) female animals. The chicken laid an egg. Turtles lay eggs which are slightly soft and leathery. This hen doesn't lay. Don’t worry, though. A lot of people I know (including myself) say lay instead of lie when referring to the action of lying down. * Lie, lie and lay have more meanings than those that I have written above. If you want to see all of them, look them up in a dictionary. I hope this could be of help to you. ―Joel

Lie (down) is also what people do on a bed and lay is its past tense. When I am tired I lie (down) on my bed, and I lay (down) on my bed yesterday because I was tired. "Lying (down)" refers to the state, and lie (down) refers to the action. Perhaps the "down" is included to avoid confusion with "lie" as in to say an untruth.