В чем разница между Press down и Press ?Пожалуйста, приведите примеры.

Most of the time, I think the difference will not be very important. Context will often be more important than which phrase you choose, in my opinion.

"press" = to put pressure on something

"press down" = to put pressure on something, with a downward motion

Most of the time when you "press" something, a downward motion is implied anyway, whether you say the word "down" or not.

"I pressed the button on the remote control" and "I pressed down on the button on the remote control" evoke almost the same mental image. If you "pressed" the button I imagine that the whole motion is complete. If you "pressed down" on the button, I might imagine that the whole motion is complete, or that *maybe* your finger was still pressing on the button. But 90% of the time, this is not important to what someone is trying to say.

I would probably write "press the button on the screen" because a button on a screen can't really physically move "down" since it is only two-dimensional. But if I read the sentence "press down on the button on the screen" I probably wouldn't think it was strange.

A time where you specifically wouldn't say "press down" is where there actually isn't a downward motion. For example, "I pressed my face against the window". This isn't a downward motion. You can even say, "I pressed my face up against the window" because it is more of an upward motion.

"I pressed on the door but it didn't open" - You wouldn't say "pressed down" here because you don't push doors in a downward motion, but a sideways motion (well, unless it is a trap door in the floor...)