Can someone explain the correct way of using lie (as in to position) vs. lay? My father always corrects me, but I never understand his description of why I'm wrong.

I have no idea if the following sentences are correct:

1)I layed next to the tree for more than an hour.

2)I went to lay down.

3)As I lay there looking at the ceiling a spider walked across my field of view.

4)I lied down in the mud.

I think they're right, but couldn't tell you why.

Help!

-Zeke