FFT is an abbreviation for a Fast Fourier Transform.
The Fourier Transform is a mathematical tool developed and named after Jean Baptiste Fourier (1768 - 1830) and is commonly used to convert a signal from the time domain (amplitude-vs-time) to the frequency domain (amplitude-vs-frequency). A frequency domain plot is also known as a spectra plot.
The Fast Fourier Transform (FFT) is a computationally efficient implementation of the Fourier Transform developed by J.W. Cooley and J.W. Tukey in 1965. The Fast Fourier Transform is limited to block sizes which are even powers of 2.
For example, if a FFT is performed on a pure sinusoidal signal, the resulting spectra would be single peak (line). Real world signals are a composite of many sinusoidal signals; examining the signal's spectra clearly shows the frequency tones which are present.