Tradition tells
us that in Archimedes' time the King was suspicious about the purity
of the gold in a crown that had been made for him by a craftsman. He
asked Archimedes to find a way to determine if it was solid gold or
a base metal covered in gold. He wanted to be sure he wasn't being cheated.

The story goes that
one day Archimedes was thinking about the problem while taking a bath.
As he lay floating in the bathtub he thought about the effect that water
had on his body, how it seemed "weightless".

Suddenly he realized
that all bodies "lose" a little weight when placed in water,
and the bigger their volume, the more weight they lose.

The weight of the
King's crown and its apparent loss of weight in water would tell him
if it were made out of pure gold.

Archimedes shouted "Eureka!" (I have found it!) and rushed out into the street naked to announce
that he had solved the problem.

Today the effect he observed is called
Archimedes' Principle and the Displacement Can we use in Physics is
called a Eureka Can!

So it was that Archimedes was credited with discovering the Archimedes
Principle that:

A body immersed
in a fluid is buoyed upward by a force equal in magnitude to the weight
of the fluid displaced by the body.

This buoyant force is a consequence of the fact that there is increasing pressure with
increasing depth in a fluid in a gravitational field.

The Archimedes Principle applies to both floating and submerged bodies and to all fluids (a fluid
is a liquid or a gas - something that 'flows'). It explains not only
the buoyancy of ships and other vessels in water but also the rise of
a balloon in the air and the apparent loss in weight of objects underwater.

Consider a cylinder
immersed in a fluid.

The pressure on the sides of the cylinder is the
same everywhere at the same depth and increases with an increase in
depth. So, points on opposite sides of the cylinder experience
equal but oppositely directed forces, cancelling each other out. As
the net horizontal force on the cylinder is zero we do not have to think
about these.

But the top and
the bottom of the cylinder are at different depths, and as pressure
increases with depth, the downward force on the top of the cylinder
is less than the upward force on the bottom of the cylinder.

So a net vertical force will act on the cylinder because of this difference
in pressure and this force will act upwards.

This upward force
is called the buoyant force.

If an object is 'weighed' with a
newton meter first in air and then when it is immersed in water, its
weight in water will be less than its weight in air because this buoyant
force will push up on it (countering the weight)

Here's an interesting one....

Further reading:

The
case of the Golden Crown - a high level
discussion of the traditional tale