A Fourier series is a way of breaking a mathematical function into a bunch of simple overlapping parts. By overlapping the parts, we can put them back together again to get the original function (or close to it). It's named after French mathematician Joseph Fourier.
To imagine how it works, think about a line. A line is just a bunch of little pieces (or parts) that are all stuck together end to end. If we take each of those pieces apart, we can make a Fourier series. So each piece is one of the overlapping parts, and when we put them all together again, we get the original line back.