If you have an understanding of Fourier transforms then you probably already have a conceptual model of transforming signals into the frequency domain. The Laplace transform provides an alternative frequency domain representation of the signal - usually referred to as the "S domain" to differentiate it from other frequency domain transforms (such as the Z transform - which is essentially a descretised equivalent of the Laplace transform).
What is the moment of a signal?
As you are no doubt aware the Laplace transform gives us a description of a signal from it's moments, similar to how the Fourier transform gives us a description from phase and amplitudes.
Broadly speaking a moment can be considered how a sample diverges from the mean value of a signal - the first moment is actually the mean, the second is the variance etc... (these are known collectively as "moments of a distribution")
Given our function F(t) we can calculate the n'th derivative at t=0 to give our n'th moment. Just as a signal can be described completely using phase and amplitude, it can be described completely by all of its derivatives.
Why is the fourier transform a special case of the laplace transform?
If we look at the bilateral laplace transform:
It should be quite apparent that a substitution $s=i\omega$ will yield the familiar Fourier transform equation:
There are some notes about this relationship (http://en.wikipedia.org/wiki/Laplace_transform#Fourier_transform) but the mathematics should be quite transparent.