They all mean the same thing in general use, but "icebox' is very old fashion, It is understood but not used much. It refers to a time before electricity when large blocks of ice were put in a box to keeps things cold. Fridge and refrigerator are identical in spoken English, though refrigerator is more likely to be used in more formal written English.

It all depends on what part of America you're in. Where I live, in Georgia, we say fridge. Up north, for example in Ohio, I have heard icebox used instead. Fridge and refrigerator are interchangeable since fridge is just a shortened version of refrigerator.

Yes, "refrigerator" is the usual word, and "fridge" is informal--not slang, but informal. "Icebox" is funny! We called the refrigerator "the icebox" when I was growing up in the 1950s, but I thought that usage had died out completely. It's old-fashioned and antiquated. Understand it when you hear it, but don't use it. Electric refrigerators for households were introduced and became common in the 1920s and 1930s. Before that, for quite a long time, people used iceboxes. These were _exactly_ what the name implies--insulated boxes the size of a small refrigerator, that were kept cold, not with machinery, but with a block of ice. Ice harvesting and ice delivery were an important industry. Ice companies cut natural ice from ponds during the winter, stored them in warehouses--insulated with sawdust--through the summer, and delivered them to customers. Regular ice delivery services were normal in all big cities.