Practical Applications and Examples of the Archimedes Principle
One example of Archimedes' Principle in action is seen in the tendency of steel ships to float, while a lump of steel will sink. Because the ship is buoyed up by a force equal to the weight of the water it displaces, distributed over a sufficiently large area, the ship floats. The key insight here is that the amount of water displaced depends on the volume of the displacing object, while the upward force depends on the weight of that water; as such, while a solid mass weighing the same as the ship would sink, the ship isn't a solid mass; the steel thus occupies a greater volume, allowing it to displace enough water to equal the weight of the ship.
Simply stated: the empty space inside the ship results in the average density of that ship being lower than the density of the surrounding water, allowing the ship to float.
The principle applies to other mediums as well. For example, air can be considered as a fluid; an unfilled balloon will sit on the ground, but a balloon filled with helium such that the average density of the balloon is less than the average density of the surrounding air will float.
As a side note, all water is not created equal! It is easier to float in salt water than fresh water, for example, because salt water is heavier than fresh water and thus the same volume of displaced water will provide a greater upward force.