Does FFT (fast fourier transform) give RMS or peak to peak amplitude vs frequency?
Peak to peak, amplitude, and RMS are all related by constant factors. However, the scope display is usually shown in dBV or similar in terms of amplitude. Note that the scope also performs windowing before calculating the FFT. I have an Agilent MSO7104A that displays its FFT in dBVrms where 0 dBV is 1 Vrms, though this may not be an industry standard.  

The FFT (fast fourier transform) is an algorithm that calculates the DFT (discrete fourier transform) which is the discrete version of the Fourier transform.  The y-axis is fundamentally the same (complex phasor (amplitude and phase) for each frequency component) but the DFT works with discrete frequencies while the FT works with continuous frequencies.  IOW, the DFT is a 'binned' version of the FT so you have a countable number of frequency bins instead of a continuous function.