A vessel that has at least three masts, all of them fully square rigged is called a full-rigged ship or just a ship. Most such ships also have a small gaff sail on their sternmost mast.
Note that usage of the term ship often leads to confusion because any large vessel is commonly regarded a ship, although strictly speaking only full-rigged ships are ships.
The full-rigged ship most commonly known or that comes to mind to those in the nautical field would be the Cutty Sark.
The full-rigged ship was a very common deep-water cargo-carrier in the 19th century. By far most of them had three masts, but many four-masted ships were built at the end of the 19th century. Many of them were later converted to four-masted barques because it was realized that a barque in most cases is a better sailer, is cheaper to maintain, and can be handled by a smaller crew.
Only one five-masted full-rigged ship was built, the Preussen.
The full-rigged ship has also been very popular for school ships because it's the most complicated type of rigging and there is a lot of work to do with all the square sails. There are a few such fine full-rigged ships that are still sailing, for example the Danish ships Georg Stage and Danmark and the Norwegian Sørlandet and Christian Radich.