The top-rated answer is actually very interesting (I had never considered Fourier space in terms of circles before) but it's less visually intuitive to me, and less obviously related to the most common applications of FT, than the way I was taught (and the way I presume it is most commonly taught), which can be seen at the top of this Wikipedia page: http://en.wikipedia.org/wiki/Fourier_series

Understanding that the square wave (and any other wave) can be represented as a superposition of sine waves at various frequencies seems like a better place to start because it illustrates the most common use case of FTs (alternating between time and frequency domain for a signal).