Understanding Fast Fourier Transforms

I have a BSEE, and my professor described Fourier Transforms (the regular kind) like this:

Imagine a recipe for making a cake.

1 cup of Flower

1 cup of Sugar

1 cup of Eggs

You combine them and you get a cake!

A Fourier Transform is like taking a sifter that sifts out only flour. You put your cake through it. How much flour do you get? 1 cup. Then you put your cake through the sugar sifter... 1 cup of sugar. Through the egg sifter... 1 cup of egg!

The ingredients are orthogonal: the flour doesn't contain any sugar or egg... the egg doesn't contain any flour or sugar.

The inputs are mapped 1-to-1 to an output. If you change any ingredients bu any amount (say 1.1 cup of flower) you get a different cake. And if you take the Fourier transform of that cake, you always get the original amount of ingredients- always

The Fourier Transform does this same thing, but with numbers on a time line, and the "ingredients" are sine waves of different frequency.

Also, not just sine waves, also cosines to account for the imaginary dimension (which will only confuse you if this is your first introduction into Fourier transforms... so you can ignore it)