Lay means "to place something down." It is something you do to something else. It is a transitive verb.

Incorrect: Lie the book on the table.

Correct: Lay the book on the table.
(It is being done to something else.)

Lie means "to recline" or "be placed." It does not act on anything or anyone else. It is an intransitive verb.

Incorrect: Lay down on the couch.

Correct: Lie down on the couch.
(It is not being done to anything else.)

The reason lay and lie are confusing is their past tenses.

The past tense of lay is laid.

The past tense of lie is lay.

Incorrect: I lay it down here yesterday.

Correct: I laid it down here yesterday.
(It is being done to something else.)

Incorrect: Last night I laid awake in bed.

Correct: Last night I lay awake in bed.
(It is not being done to anything else.)

The past participle of lie is lain. The past participle of lay is like the past tense, laid.

Examples: I could have lain in bed all day.

They have laid an average of 500 feet of sewer line a day.

Layed is a misspelling and does not exist. Use laid.