The Fourier Transform, explained in one sentence (2014)
Ok, my knowledge of the Fourier transform is limited to the discrete version, but I might as well add my take into the mix. :-)  I've always thought of the Fourier transform as a series of projections (as in high dimensional dot products). You dot product your signal with a set of sine and cosine waves, which are selected to form an orthogonal basis over the domain of the signal. So it's just a basis transformation. The phase of the wave is captured by the fact that a sine and a cosine  of the same frequency, when added together, will always remain a sine function, but with a different phase, depending on the relative amounts of sine and cosine that are added together.