The verb "lie" is an intransitive verb which means "to recline" or "to be situated". The present participle of "lie" is "lying," the past form is "lay" and the past participle is "lain":

The cup is lying on the floor.

The cat lay in the sun all morning.

The newspapers had lain on the verandah for two weeks before anyone noticed that Mr. Gilfillian had disappeared.

In each of these examples, the intransitive verb "lie" is used (in conjunction with an adverbial phrase) to describe the location of the subject.

The verb "lay" is a transitive verb which means "to place" or "to put." The present participle of "lay" is "laying," and both the past form and the past participle is "laid":

I was laying the cups and saucers on the table when I dropped one.

Jenkins laid the suspicious parcel on the commissioner's desk.

The supervisor had laid a cup of scalding coffee on the counter only moments before the bulldozer rammed into the construction office.

In each of these sentences, the transitive verb "lay" is used to describe the fact that someone had placed something somewhere.