Lay: "To cause to lie" (OED v.1)

Lie: "To be in a prostrate or recumbent position." (OED v.1)

The two terms are extremely similar and one involves the other, though they have different uses. One can not lay on the floor themself; lay is something that they do to something else. In contrast, lie is used when the action is being done to oneself. 

Cow

This cow is resting on the field. Since the cow is performing the action upon itself, it is lying on the field, as opposed to laying.

Example 1

Incorrect: I want to go lay down.    

Correct: I want to go lie down. 

Explanation: Using lay in the present tense requires a direct object separate from the subject, and "I" in the sentence is the subject, so lie is the correct word here.

Example 2

Incorrect: Sam and I will lie down our tools.  

Correct: Sam and I will lay down our tools..

Explanation: The tools are the direct object in the sentence, and the subjects are "Sam and I", therefore we should use lay to reference the direct objects.

Example 3

Incorrect: Vicki tells her dog to "roll over," and then "lay down" for a treat.

Correct: Vicki tells her dog to "roll over," and then "lie down" for a treat.

Explanation: What's the difference? Lay is used when there is a subject and separate direct involved. In this example, the dog is both the subject and the direct object of the action. That is why lie is used in the correct example.

Khan Academy Lesson

Lay is used when the action is done by the subject to the direct object.

Lie is used when the subject is the direct object.

Meme Generator

"Lay vs Lie | Ask Linda! | English Grammar".