Think of any little layer of the water. It has some weight. It would fall unless there were some force pushing it up, canceling the gravitational weight force. That means that the water beneath it must be pushing up a little more than the water above it is pushing down. In other words, the deeper you go, the higher the pressure must get.
In practice, water, like many other fluids, is only very slightly compressible. The deeper water is at higher pressure, but that corresponds to only a small increase in its density.
Anyway, if you place an object in water that weighs the same amount as the water it displaces, the pressure and weight forces on it will just cancel, the same as they would for the water it displaced. If it's heavier than the water it displaces, then the weight force wins and it will sink. If it's lighter, the pressure difference wins and it floats up to the surface.
So whether an object floats in water depends on whether it weighs more or less than the same volume of water. Weight density means weight per volume, so this is the same as asking whether it is more or less dense than water.
(published on 04/10/10)