Fourier Transform is the general name of the technique, labeling that it is a Transform and marking which one it is.
Fourier Series is the formal ddefinition of the mathematics of the transformation.
The Fourier Series is an infinite series in most cases, and so often requires an infinite amount of memory except in situations where you can write mathematical formulas for the terms. However in most cases you do not know the mathematical formulas for the terms when you have arbitrary input.
fft, Fast Fourier Transform, is an adaptation to finite discrete signals and so is suitable for arbitrary input of unknown properties.
