Lie vs lay. As the present tense "Lie down" or infinitive "I want to lie down."
It's incorrect™ to say "Lay down here" or "I want to lay down" because present-tense "lay" is something you do to something else (transitive) - "I'll lay the blanket on the grass." "The hen lays an egg."
There is widespread and persistent confusion though, because "lay" is also the past of "lie" - "I lay in bed for hours this morning." And "lying" ("I am lying on the floor") is v close to "laying" ("The decorators were laying the carpet."
Now.
It seem to me that the misuse of "lay" is becoming more and more widespread (am I right?) and that it's especially common in America (am I right?) to the extent that it's on the verge of becoming correct or at least accepted as "English as she is spoke".
Dictionaries have been of no use and a web search simply turns lots of grammarist sites that would alienate the kind of person who say lays down faster than a gay Muslim Mexican immigrant single mother.