A ship is sunk when every one of its cells has been hit. The game log names the ship the moment its last cell goes down.
A ship sinks when all of its cells have been hit. A ship that has taken some hits but not all is damaged and still afloat.
What you are told
After each shot you see whether it hit or missed. When your shot hits the last remaining cell of a ship, the game log says you sank it and names the ship, for example the Carrier or the Destroyer, and the ship is marked as sunk on the enemy grid.
Why it is useful
Knowing which ships are gone tells you which are still hiding, and so how big the gaps you are searching can be (the five ships and their sizes). If a line of hits is longer than the ship you just sank, two ships are touching, so go back and finish the other. See the strategy guide and the full rules.