There is no typical type of vessel that can be called ”a pirate ship”. The most pirates got their ships by capturing them. Besides, a lot of them acquired ships in a mutiny. It was also a common thing that legal ship owners, like the merchants and the privateers, sometimes turned to piracy. However, there were other unusual examples, like Steed Bonnet's one. He bought a ship from his own pocket.
When the pirates did not find a ship useful, they sunk or sold them. After they obtain a ship, the pirates usually adapted it for their specifications. Generally, they used various types of ships, from small, one-masted vessels to the most powerful warships, as they performed numerous different jobs.
Probably the most important thing to the pirates was that ship is fast, light and maneuverable because they wanted to catch-up target as soon as possible. Besides, they sometimes needed to escape quickly from well-armed warships and pirate hunters. That is why a shallow draft was also preferred. With that advantage, the pirates could easily flee from large powerful vessels by hiding in shallow waters where pursuers could not follow them. The perfect pirate ship would also include many mounted cannons.
A pirate ship should be small, easy to careen and repair, but with a lot of working space and large living quarters for the usually crowded crew and a big storage for plundered goods. Judging from this description, perfect match for the pirates would probably be a sloop or a schooner.