When you ask the guy behind the counter if they've got any aspirin, you're making an inquiry. Almost any search for information or knowledge is an inquiry, though an inquiry is often an official search.
Though any question is, technically, an inquiry, that word is usually used to refer to an official or public search for the truth. For instance, after a plane crash, the government launches an inquiry into the cause. Politicians and government officials are often the ones who demand an inquiry when an important question needs to be answered, but a child can do the same thing. If your kid wants to know what's for dinner, she can make an inquiry about it. She can also just, y'know, ask.