Navigation
Navigation is one of five duty puzzles. (The others are sailing, carpentry, bilging, and gunning). This puzzle is reserved for officers in the crew that owns the ship. Other players my also be ordered to navigate by such an officer. There are two 'flavors' of navigation, usually called Duty Navigation (duty nav or dnav for short) and Battle Navigation (or bnav).
Duty Navigation multiplies the effectiveness of your ship's sailors, and allows memorization of ocean routes.
Navigating involves lining up specific patterns of pieces on a circular board. There are three rings, each with eight positions around them. (up, down, left, right, and diagonals) Somewhat random stars fall towards the middle, starting on the outside. You can contol the three rings, selecting any one and rotating it (and the stars it contains) left or right. Stars with nothing "underneath" them will fall towards the center. Lines of three or more stars, either from the center out or along a ring, will disappear. The main goal is to line up appropriately colored stars in the marked positions, forming a "constellation" which will then disappear and add to your score. Simply lining up same-colored stars doesn't do much to increase your score.

The key to navigation, since you don't know what constellation is coming next, is simply to be as efficient as possible. It's also a good idea to keep multiple colors of stars on any level as it starts to fill up, both in case the next constellation needs it, and so that you have multiple options if your board begins filling up and you need to line up stars to free some space.
Most of the time, you want to fill constellations from the center out. First correctly arrange the stars on the inner ring, then the middle, and finally set up any stars on the outer ring. An exception is when you get the rare constellation like the one pictured above, where the constellation contains a line of three like stars, in which case you have to finish that part last.
When you first start learning duty nav, you will get bad scores (Poor or even Booched) because the scores are strongly weighted on the number of stars in the constellation you completed. The constellations in the first couple levels have only one or two stars and hence score very low, even if you complete them in the fewest possible stars. Try to practice this puzzle in the navy to build up skill (you will have to have Broad experience in the other four ship duty puzzles before you get these mission).
This is another term for the sea battle puzzle, where a commanding officer controls the movement and firing of guns in a live battle with one or more other ships (also see blockade). The player who most recently selected Navigate from the ship's wheel is automatically put in charge when a battle begins. Others may take the helm by selecting navigate from the wheel. This switches control to the new player.