What is Archimedes' Principle?

In short

Archimedes' principle says that an object placed in a fluid feels an upward force, called the buoyant force, equal to the weight of the fluid it pushes aside.

In simple words

The buoyant force is $F_B = \rho_{fluid} V_{displaced} g$ . An object floats when this force can equal its weight, which happens when its average density is less than the density of the fluid. A steel ship floats because its hollow shape displaces a large volume of water. A solid steel block sinks because its density is greater than water's.

Example

A block of volume $0.0020\ \text{m}^3$ is fully under water of density $1000\ \text{kg/m}^3$ . The buoyant force is $1000\ \text{kg/m}^3 \times 0.0020\ \text{m}^3 \times 9.8\ \text{m/s}^2 = 19.6\ \text{N}$ .