There are actually quite a few meanings for these words, but the most common are below.
1. to lie (regular verb lie, lied, lied) = to deliberately say something untrue
Johnny lied to his teacher about his homework.
2. to lie (irregular verb lie, lay, lain) = to be in or get into a flat position
I feel sick. May I lie on your sofa?
I felt sick so I lay down for an hour.
Yesterday I lay on the grass for an hour.
3. to lay (irregular verb lay, laid, laid) = to put something down, often flat
Please lay the blanket over the baby.
4. to lay (irregular verb lay, laid, laid) = [of a female bird etc] to produce eggs
This chicken laid 6 eggs this morning.
My American dictionaries suggest that the above is correct usage in American English, but I hear many Americans using to lay for 2 above, thus:
I feel sick. May I lay on your sofa?
I conclude that this is also correct usage today in American English but perhaps some of our AmE speakers could confirm.