frequency folding due to aliasing in FFT
The Fourier transform $X(e^{j\Omega})$ of a sequence $x[n]$ is by definition periodic of period $2\pi$ in $\Omega$. The $N$-point discrete Fourier transform (DFT) of $x[n]$ is simply defined as a sampled version of the Fourier transform $X(e^{j\Omega})$, that is
$$X[k] = X(e^{j\frac{2k\pi}{N}}).$$
The periodicity of $2\pi$ in $\Omega$ for the Fourier transform becomes a periodicity of $N$ in $k$ for the DFT
$$X[k+N] = X[k].$$
The negative frequencies showing up to the right of the positive frequencies is simply a consequence of this periodicity. Or if you prefer, you get $X[k]$ for $k \in [0, N-1]$ instead of $k \in [-\frac{N-1}{2}, \frac{N-1}{2}]$. If you want the vector to contain first negative frequencies, then the DC and finally the positive frequencies instead, you can for example use MATLAB fftshift.

Those negative frequencies are not shown for the "real" DFT because if $x[n]$ is real, then $X[k] = X^*[-k]$ (conjugate symmetry). Hence, you only need $X[k]$ for $k \in [0, N/2]$.