If a machine is push-button, you work it by pressing a button with your finger, rather than turning a handle, turning a key, or winding something. A push-button phone, a push-button lock, push-button controls. The hyphen is usual when the two parts sit in front of a noun. You may also meet push-button as a name for the button itself, though button already does that job. Though actually, people also use it as a picture-word. A “push-button war” or a “push-button world” means something you can start or control too easily, as if one finger were enough and you never had to get your hands dirty. That use is a little critical: it says the action has become too simple, and maybe too careless. The home meaning is still the physical one: press, and it works.