The Fourier transform is used to take a function into its reciprocal space. The reciprocal space is used to describe what component of a function behaves with a certain periodicity. Like, suppose you are at the ocean, where waves are coming in with spacing of 20 meters. There is a function describing the height of the water at each point; this function varies everywhere. Your reciprocal space water height function is simpler, for it would be zero everywhere except one high area at 2π/ twenty meters. Just looking at it, you would say, "Oh, that's one cycle per twenty meters!"
Secondly, the process of Fourier transformation upon a function is nearly self-inverting. That is to say, the formula for taking the fourier transform of a function is the complex conjugate of the formula for taking the inverse fourier transform. This is true, at least, up to a constant factor for any valid Fourier transform; the form used above does not have this property, because the fourier transform and its inverse differ by a factor of 2π. However, one can choose both to be the geometric mean, in which case they are again complex conjugates.
Thirdly, the fourier transform of a product of functions f(x)g(x) = the convolution of their individual fourier transforms F(k)oG(k). Applying the principle from earlier, we can reverse this to Fourier transform of F(k)G(k) = f(x)og(x). This is a very useful identity because the convolution comes up in a wide variety of problems, and can be very difficult to calculate otherwise.
One nifty way to think of the Fourier transform is as a change of basis in function space from dirac delta functions to complex corkscrews.
The only nontrivial function which is preserved under Fourier transform is the Gaussian curve.
If the function is periodic, then all components that are not multiples of this wavenumber are 0 - it is a series of spikes. This series is called a Fourier series. If you know in advance that the function is periodic and you know the period, you can save effort by only computing the elements of this series.