LIE and LAY is a common grammar issue. We actually teach a unit on this in our Analytical Grammar program.
The difference between LIE vs LAY is important. It has to do with whether or not your verb is transitive or intransitive. The root is TRANS, so imagine a truck TRANSPORTING something from the subject. A transitive verb takes a direct object, so there is cargo on the truck. An intransitive verb can’t take a direct object, so that truck will not transport anything.
The problem with LIE and LAY arises because the words cross over each other. See in the graphic below how LAY is in both lists? That becomes a problem for many. So here we go:
LAY: to put down; needs an object (transitive)
LIE: to recline; never takes an object (intransitive)
For example, you might LAY a book on the table, LAY a sweater on the bed, or LAY a child in her cradle. When you feel tired at the end of the day, you may LIE down. But you can’t “lie” a book anywhere, and you can’t “lay” down (no object) at the end of the day.
*The past tense of lie is LAY, so:
The cat lay in the mud after it rained yesterday.
*The past tense of lay is LAID, so:
Last week, I laid the book report on your desk.
*The past participle of lie is LAIN, so:
The pig has lain in the mud for hours.
*The past participle of lay is LAID, so:
I have laid my clothes for the party on my bed.
HOPE THIS HELPS!