Why do heavy ships float but a small rock sinks?

The key is that floating depends on density and displacement, not on total weight, which is why an enormous steel ship floats while a tiny pebble sinks. When any object sits in water, it pushes aside (displaces) some water, and the water pushes back up with a buoyant force equal to the weight of the water displaced (this is Archimedes principle). If an object displaces water weighing more than the object itself, it floats; if it displaces water weighing less than itself, it sinks. A small rock is solid and denser than water, so it can only displace a small amount of water, weighing less than the rock, and it sinks. Weight alone is not the issue, it is how much water you shove aside compared to your own weight.

A ship, despite being made of heavy steel and weighing thousands of tons, is mostly hollow, a huge shell full of air. That shape lets it displace a massive volume of water, and because so much of the ship's interior is empty space, the ship's overall average density (the steel plus all the air inside, spread over its whole volume) is less than the density of water. The ship sinks into the water only until the weight of the water it has pushed aside equals the ship's total weight, and then it floats, riding at that level. This is why the same lump of steel behaves differently depending on shape: a solid steel ball sinks (dense, displaces little), but hammer that same steel into a wide hollow bowl and it floats (now it displaces a lot of water for its weight). It is also why an overloaded boat rides lower and can eventually sink, adding cargo increases the weight until the hull can no longer displace enough water to hold it up. So heavy ships float and small rocks sink for the same underlying reason: floating is a contest between an object's weight and the weight of the water it displaces, won by being less dense on average than water, which a hollow ship achieves and a solid rock cannot.