For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that
$$X_{N-k} = X_k^*$$
where * denotes complex conjugation.
As an example of the consequences of this, consider the function $f(t) = \sin(20*2*\pi*t)$. This could be thought of as a signal oscillating at a frequency of $20$ Hz. If I were to sample this signal at a rate of $100$ Hz, and then DFT these samples into Fourier coefficients, I would find a non-zero coefficient corresponding to $20$ Hz and another coefficient of equal magnitude corresponding to $80$ Hz (and zero everywhere else).
Is there a non-mathematical reason to expect this? Is there any meaning, physical or otherwise, that can be given to the coefficient corresponding to $80$ Hz? Is there some property of the input signal that oscillates at $80$ Hz, or some interpretation that would make it seem to be, or is it just a mathematical artifact?