In music, see below.
Intensity is a measure of the time averaged Energy Flux. To find the intensity, take the energy density (that is, the energy per unit volume) and multiply it by the Velocity at which the energy is moving. The resulting Vector has the units of Power divided by Area (i.e. Watt/m²).
Anything that can carry energy can have an intensity associated with it (i.e. it is possible to define the intensity of the water coming from a garden sprinkler), but intensity is used most frequently with Waves (i.e. Sound or Light).
If a point source is radiating energy in three dimensions, and there is no energy lost to the medium, then the intensity drops off as the distance from the object squared. The reason why this is so is one part physics, one part geometry. The physics comes from the conservation of energy (i.e. if the energy isn't being lost to the medium, it can't just disappear). The consequence of this is that the net power coming from the source must be constant, thus:
P = ∫I·dA
where P is the net power radiated, I is the intensity as a function of position, and dA is a differential element of a closed surface that contains the source. That P is a constant. If the source is radiating uniformly, that is, the same in all directions, and we take A to be a sphere centered on the source (so that I will be constant on it's surface), the equation becomes:
P = |I|(4πr²)
Where I is the intensity at the surface of the sphere, and r is the radius of the sphere (note: enclosed in parentheses is the expression for the surface area of a sphere). Solving for I, we get:
|I| = P (4πr²)-1
If the medium is damped (i.e. both sound and light in air slowly lose energy), then the intensity drops off more quickly than the above equation suggests.
In music, intensity is the perceived loudness or volume of a sound. It is often cited as one of the fundamental aspects of music.