Fourier transform terms explained

I know there are lots of tutorial on different webpages.. but I am not an engineering student... those look quite complex to me..

Could anybody explain what is "Fourier transform" in the very simple sentences so non-engineer background people can actually understand?

Thanks...

• A transform, used by Joseph Fourier. Dec 22 '12 at 2:32
• @Ethan It is a pity I cannot down vote comments like that. If you have something that might be useful, then say it otherwise don't.
@Quiaochu's answer is spot-on for temporal signals. In optics, however, there is also a spatial interpretation as follows. Consider some wavefront $f(x)$ that may be propagated along the $x$ axis. For example, the wavefront may be a focused, circular wave, or a diverging wave, or even some aberrated wavefront. The Fourier transform allows us to express a complex wavefront in terms of a sum over plane waves that propagate at a range of angles with respect to the $x$ axis. In this case, the Fourier transform $F(v)$ represents the amplitude of the plane wave whose angle $\theta$ of propagation with respect to the $x$ axis is given by $\frac{2 \pi}{\lambda} \cos{\theta}=v$, where $\lambda$ is the wavelength of the light. This is known as the angular spectrum representation of a wavefront, and this representation allows for beautiful analysis of wave propagation in complex optical systems.