Don’t Say: 
– Please lie the book on the table.
– I’m going to lay down in bed.

Say This:
– Please lay the book on the table.
– I’m going to lie down in bed.

Remember:

• You lay something down.   I’m going to lay the baby down for a nap (laying him down – not lying down yourself).  My daughters lays her clothes on the bed each morning.
• People lie down by themselves. Every night I fall asleep as soon as I lie down in bed.  He lies down on the sofa to watch TV at night.
• Lie – not lay – is also used when you mean saying something that is not true.  Don’t lie to me.

Past Tense of Lie / Lay (This gets confusing)

• The past tense (and past participle) of lie (as something not true) is lied. She lied to her husband about the cost of the shoes she bought.  She has lied about this many times.
• The past tense of lie (meaning to recline) is lay.  He was sick, so he lay in bed all day yesterday.
  Many native English speakers use laid instead of lay. (He laid in bed all day.)
• The past tense of lay is laid.  She laid her earrings down on the night table.  The chicken laid 3 eggs.

• More detail on this from Grammar Girl.
• A short YouTube Lesson
• And a Quiz (short, but difficult – even for native English speakers!)

— Thanks to Carrie for suggesting this article —