What does the word 'bin' mean in the context of FFT's?
A finite length DFT transform has a finite number of basis vectors, which are all single frequency complex sinusoids.  If the results weren't binned then you would need an infinitely long DFT or FFT result to represent any continuous spectrum.  However, Parseval's theorem says that all the energy in a finite input vector ends up in the finite FFT result vector.  So any single between-bin-center-in-frequency pure sinusoid (or between DFT basis vector frequency, windowed by any finite length FFT) ends up binned, spread mostly into FFT result bins very near in frequency.

"very near" because:

Each FFT result bin is not rectangular, but a filter in the shape of Sinc function (more precisely, a periodic Sinc function or Dirichlet kernel), since that is how a pure sinusoid correlates with any one DFT complex basis vector when swept across the frequency range from 0 to Fs/2.  The result is also conjugate mirrored in the full length FFT result for strictly real input.

Another interpretation is that there exist no in-between bin frequencies, that any cut-off sinusoid is really a repetitively discontinuous periodic waveform.  But that mythical input is not representative of how FFTs are actually used in many forms of DSP signal analysis (windowing, peak estimation, low offset overlapping windows, phase vocoders, etc.). Although knowing about this circularity is important when using an FFT for fast convolution.