You can understand the basics of what it means and how to use it for some practical applications with very little maths (i.e. high school level), if you are happy to use computer software to crunch the numbers for you.
At the other end of the scale, Springer publish the Journal of Fourier Analysis and Applications, for new research papers.
A random selection of applications for it are image processing, signal processing, data compression (e.g. MP3 audio and and JPG video), and advanced methods for solving ODEs and PDEs.
All undergrad electrical engineers take a course (or set of courses) on "signals and systems" that is essentially applied Fourier analysis, both in discrete time, continuous time, etc., along with related tools like Laplace and Z transforms. You will find it used in a large percentage of EE disciplines - signal processing, communications, electromagnetics, etc. It is hard to underestimate its importance for EE. It is also a lot of fun. I use Fourier analysis almost every day in my work (I am an EE). My EE courses carefully stated all the convergence theorems, but did not prove them; at that level all you really need is calculus to understand Fourier. Proving the convergence theorems is another story altogether, though.
edit: linear algebra is also helpful for understanding Fourier - it was a prereq. for our signals class and the ideas from linear algebra are natural to use to think about fourier series, both in continuous and discrete time.
I can echo Fourier Analysis' importance in EE. I use Fourier concepts daily in my work as well. While many EEs don't do a lot of hand analysis, we have to interpret a lot of information in the frequency domain and that is where deep understanding of the concepts of Fourier Analysis is important.
Basically, Fourier showed that all periodic functions can be broken down into a sum of simple sinusoidal waves of various frequencies. (Think superposition principle, if you know it) The Fourier transform takes a function and gives you the coefficients of the terms in that sum.
An application is filtering. A noisy signal looks like a clean signal, but has lots of tiny spikes and troughs on it. Using Fourier analysis, we can find the coefficients for the sum, then set them equal to zero beyond a certain frequency threshold. Then the tiny spikes-which correspond to very high frequencies- aren't added into our sum, so when we put everything back together we get a nice smooth and clear signal.