Let's just go ahead and get this one out the way. You knew Sid Meier's Pirates! would be on this list somewhere -- and rightly so. The 2004 remake of the 1987 version is as captivating a pirate game as any, and it's arguably the best of the lot. Developed by Firaxis Games (the makers of both the XCOM and Civilization series), this game sets you loose in the 17th century Caribbean after the nefarious Marquis de la Montalbán kidnaps and enslaves your family. Although you are nudged throughout the game to track down your family members and defeat Montalbán, the real enemy in this game is time.
As time passes, your character ages and will eventually be forced to retire, so the decisions you make on how to seek your fame and fortune are pretty important. Do you want to find the perfect trade route to haul spice and make an honest living? (But why would you?) Will you fight for your nation's navy and contribute to the war effort, conquering enemy ports and towns in the name of the various Governors you'll come to meet? Or will you go all in on the full pirate experience and become a legend, flying the colors of the Jolly Roger?
Regardless of how you play, it's a good time, and with Sid Meier's Pirates! being on nearly every platform now, you have no excuse not to pick this one up. Just don't pirate it.