After you've completed the 'The Merryweather Heist' main storyline mission you can purchase the Sonar Collections Dock on the west coast (which unlocks a sweet submarine) for $250000. A character named Abigail shows up on the dock after you've purchased the property. Approach her (as Michael) and a new mission starts; you'll have to collect 30 submarine pieces to complete it.
Starting this mission automatically unlocks a much-needed Dinghy at the dock, a speedy boat which comes (like the Submarine) with diving gear when you exit the boat while in the water. We should tell you up front that the reward for this quest is terrible ($10 - no joke), so you should view the Dinghy as your reward, along with the bragging rights of having actually done this mission. It doesn't count towards 100% game completion either, so it's up to you. Even though the mission itself isn't very rewarding, the areas in which you're looking for Submarine Pieces often contain Hidden Packages, which are roughly $10000 worth a piece (see section [HPL]), so you could view those as your reward.
Finding the pieces is very easy; as soon as you enter the Dinghy a green circle shows up on your map (you'll want to take a look at the main map). Enter it and a sonar shows you exactly where to find all pieces in that area. Clear it and move on to the next region, again marked with a green circle on your map.