What exactly is Archimedes' principle

Archimedes was a Greek mathematician and scientist. He was born in the Greece city-state of Syracuse in 287 BC. A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Archimedes' inventions include a proof of the principle of the lever, the widespread use of the concept of center of gravity, and the explanation of the law of buoyancy. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

If you put stone into a water container, volume of water displayed by stone is equal to volume of stone itself. When students are taught about Archimedes principle in class 8th or 9th they sometimes mistaken only above part as Archimedes principle but there is more to it. Lets look at more exact statement of the principle

"The upward buoyant force that is exerted on a body immersed in a fluid, is equal to the weight of the fluid that the body displaces".

Consider an experiment in which a stone of mass M attached to a sting is dropped into water. Let volume of stone be V which is also equal to volume of water displaced, density of stone ρ_s and that of water is ρ_w. Then according to Archimedes principle, 

Upward force on stone also called as buoyant force = weight of water displaced = ρ_wVg, 

while downward force on stone = Gravitational force = Mg = ρ_sVg.

Therefore net downward force on stone F = ρ_sVg - ρ_wVg = (ρ_s - ρ_w)Vg. 

This results into three cases depending on relative value between density of stone i.e. 

If ρ_s > ρ_w stone will sink into the water since F is positive. 

If ρ_s = ρ_w  stone will neither sink nor float since F=0. 

If ρ_s < ρ_w  stone will float into the water since F is negative

Actually, the origin of buoyant force can be understood as when stone wasn't there, water in that portion was prevented from falling down due to gravity by water below it . So to maintain the portion in equilibrium upward force equal to  ρ_wVg was always present and even after water was replaced by stone, there isn't any different treatment to it i.e. same upward force was applied on the stone.

Archimedes principle is useful measuring volume and density of irregular objects. It describes working of large ships and submarines.