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