The basic idea of Fourier analysis is that every "reasonable" function on a [0, 2*\pi] can be analyzed as the superposition of an infinite number of sine and cosine waves with different frequencies. A whole field of mathematics, called harmonic analysis, developed from this beautiful idea.
In this course you will learn the basic Fourier series expansion of periodic functions. Later we will use the Fourier series expansion to solve initial and boundary value problems of partial differential equations.
The "spectrum" of periodic functions are discrete, and the main point of Fourier series is that every periodic function is the sum of an infinite number of simpler functions, each of these simpler functions has "exactly one frequency". Allowing the period to go to infinity, one gets all functions on the real line. But when one wants to reconstruct a function on the real line from single frequency functions, one needs to a "continuous sum", namely an integral. This is what Fourier transform is about. Since the theory of Fourier transforms is more involved than that of the Fourier series, we will only be able to scratch the surface in this course.