Archimedes's principle states that the buoyant force acting on an object immersed or floating in a fluid equals the weight of the fluid displaced. What does that mean?

First what is a fluid? Ordinarily most people think of a fluid as a liquid. The scientific definition of a fluid however includes either a liquid or a gas. Any liquid or gas is a fluid.

The buoyant force is the upward force on an object immersed in or floating on a fluid. If the object weighs less than the buoyant force, it will float on top of the fluid. If it weighs more, it will sink.

Because the buoyant force equals the weight of the fluid displaced, an object must displace a greater weight of fluid than its own weight in order to float. That means that to float an object must have a density (mass divided by volume) less than the density of the fluid. If the object's density is greater than that of the fluid, it will sink.

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Archimedes' principle does not consider the surface tension (capillarity) acting on the body[4], but this additional force modifies only the amount of fluid displaced, so the principle that Buoyancy = weight of displaced fluid remains valid.

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyancy force on an object is going to be equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyancy force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

http://en.wikipedia.org/wiki/Buoyancy#Ar...
Archimedes principle explains how buoyancy can be used to determine density.