There is no single answer to this, as people use the terms differently in different contexts. However there are some general guidelines. A battleship, from "line of battle ship", is a big, heavily armoured ship with a lot of guns, intended to be able to fight anything that floats. That's also a good description of a dreadnought. The difference is the "HMS Dreadnought", a ship built in 1905 in England; it marked a departure to a considerably different design philosophy. Earlier battleships had had a mix of gun sizes for their main armament; for example, the USS Texas, the first battleship of the US Navy, had 2 twelve-inch and 6 six-inch guns. HMS Dreadnought had 10 12-inch guns. (Both also had a bunch of small-calibre quick-firing guns for close-in defense, and some torpedo tubes; but we're talking about main armament.) Additionally, the Dreadnought had turbine engines fueled with oil, making it faster than previous battleships even though it had more displacement. In other words, it was more heavily armoured, more heavily armed, and faster than any other battleship in the world. It was seen as such a step forward in combat power that from then on, basically all battleships were built to the all-big-gun philosophy, and older battleships got the adjective "pre-Dreadnought". So if this is the context, then a dreadnought is a battleship built after 1905 or so, or alternatively, it is a battleship whose main armament is all one large calibre and with oil-fired turbine engines.

However, as time went on, the older battleships were retired, and eventually people used 'battleship' and 'dreadnought' interchangeably because that was basically the only kind of battleship there was. (Of course you can find exceptions to this rule, especially in the minor powers; for example the Norwegian coast-defense battleships Eidsvold and Norge, sunk by the Germans in 1940, had been considered quite powerful when they were built in 1899.) You sometimes see 'dreadnought' used to mean 'especially powerful battleship', especially perhaps in military science fiction with space navies, where the distinction is usually one of size - dreadnoughts and battleships are just different sizes of the same kind of ship. In that context you may also sometimes see 'super-dreadnought' to mean an even bigger ship, still of the same kind; to the best of my knowledge this has never referred to any class of ship in Earth history.