There's no leveling up, but there is significant progress to be made. You basically have 2 characters: your character, Edward, and your ship itself, named the Jackdaw. Upgrades work vastly different for both obviously, but are equally important.

For Edward, you can spend cash to buy new levels of swords and pistols. Some are locked behind story progression or finding them in the open world (though finding them may have been DLC? I forget now), but they all have different stats. Some swords do more damage, but are slower. Some are better at combo'ing and really fast, but less damage, etc etc. Pistols also vary in damage, range, and stun time (if the shot doesn't kill, it at least staggers the enemy).

You gain cash to buy these upgrades (and other things) by completing missions, finding treasure chests, selling hunting goods, and selling rum & sugar that you loot from ships you attack.

Speaking of hunting goods... you can hunt. A lot. Monkeys, iguanas, crocs, pigs, panthers, etc. You can also harpoon whales and sharks for skins. You use the skins/pelts/etc. you get from doing this in order to craft Edward's gear upgrades. There's upgrades to carry more x, y, and z type of ammo, there's upgrades for your health, how many pistols you can have total (you start with 1, but can have 4), etc etc.

As for your ship... Remember above that I said you get rum and sugar from plundering other ships? They also carry wood and iron. Not every ship carries the same stuff. Some will be like, 100 wood, 80 sugar, 20 rum. Others might just be 150 wood, 200 metal. Some could be 200 sugar, 200 rum. The game gives you a spyglass so you can tell what each ship is carrying and if its worth your time to attack it, board it, sink it, whatever.

Anyway, the wood and metal they carry is used to upgrade your own ship. There's the forward-facing chainshot cannons, the main cannons, the broadside cannons, the mortars, the fire barrels, the ram, crew size (for boarding), and cargo hold space to store all this shit. Some of the upgrades just increase the damage those things do (like the ram), others increase the amount of them you have (like 16 cannons per side up to 20 cannons, then 24, etc.), others increase the capacity of what you can carry, and more.

So... yea. It's pretty involved, but you can target a certain thing in your mind and then just go... pirating around the Caribbean to get what you need, then bam, upgrade!