Sail the Caribbean in search of fame and fortune
In Pirates you start the game in Port Royal (in Jamaica) with a modest amount of money and a small ship. Officially you're a merchant. You can be peaceful and seek a steady profit by trading around the Caribbean. Run to Laguna de Terminos for a cargo of logwood, stop in Hispaniola for provisions and sell at a profit in Port Royal.
If you carry your cargo to Europe you'll find better prices and make a lot more money, but the Atlantic is risky unless you can invest in a larger vessel.
A hazard of trading peacefully is that the Spanish think the whole of the new world (except for a bit of Brazil) belongs to them. The Pope told them so - although what it's got to do with him is anyone's guess. So the Spanish will treat you as a pirate even if you're not. And you can't actually act like one, unless you are. You can fight them off, but you can't loot and plunder their ships, or you risk being arrested when you get back to port.
If you don't care for being chased around the by the Caribbean by the Spanish then you can fight back. In Port Royal you may be offered the chance to purchase a "Letter of Marque", which is a licence to attack Spanish ships and possessions.
This makes you a privateer rather than a pirate, and it means you can take captured ships and cargo into Port Royal and sell them. Legally. It's only legal with the English and French, but the Spanish are calling you a pirate anyway - so who cares how they reckon things?
As a privateer you're only allowed to attack the Spanish. You can't touch ships belonging to other countries, and as the game goes along you'll find fewer of the ships you meet are Spaniards (the more of them you capture and sink, the fewer will be left).
As a pirate you can attack anyone and anything. If you think you're tough enough, that is. The downside is it won't just be the Spanish navy you have to look out for (not that you're afraid of them, of course). If you're a pirate the Royal Navy will be looking for you as well. There are more of them, in better ships with better crews. They fight harder, so you'd better have a fast ship. Running away is likely to work better than fighting.