Lay (laying, laid, laid) = put something somewhere: ‘She laid the money on the table in front of me.’
Lie (lying, lay, lain) = be in a horizontal position (on the ground/on a bed etc.) ‘He was lying on his back, staring up at the sky.’ ‘I could lie here all the day!’
Lay down= put something down: ‘Just lay the books down on the floor.’
Lie down= put your body in a horizontal position, especially on a bed: ‘I think I will go and lie down and lie down for half an hour.’