First, you'll need a fishing rod, which can be obtained from a fishing master. After that, go to one of the fishing spots on the dock on either Port Royal, Tortuga or Padres del Fuego. Throw your lure in the water (click once to set throw power, click again to throw). Let the lure hang in front of a fish, and wait for a bite. Once a fish is on your line, hold the left mouse button to reel it in. When a red ! appears, release the button, or the fish will snap your line. Reel the fish in all the way to catch it.
As you catch more fish, you will gain experience and eventually level up your fishing. Doing this, you will unlock new fishing skills, which can open new and more effective strategies for fishing. Use them wisely.
With the basic rod, you can only catch the smaller fish near the surface. Your fishing rod can be upgraded for gold at level 5 and level 15. Once you upgrade your rod once, you will be able to reach deeper seas and catch medium size, and finally the largest fish with the second upgrade.
When catching the largest fish, the red ! doesn't appear, so you'll have to look out for the fish's movements instead.
Bamboo poles with hooks covered in bait (to misinform the fish of your intent)
See also: How do you catch Legendary Fish