The Discrete-Time Fourier Series finds the frequency representation of a signal that is both periodic and discrete in time. We derive the Discrete-Time Fourier Series much like we derive the Discrete-Time Fourier Transform -- by sampling our periodic signal $x(t)$ and plugging it into the Fourier Series equations.
The result is a pair of direct and inverse discrete-time Fourier series is defined by $$ X[k] = \frac{1}{N_0} \sum_{n = (N_0 \textrm{ period})} x[n] e^{-j n \frac{2 \pi}{N_0}k }$$ $$ x[n] = \sum_{k = (N_0 \textrm{ period})} X[k] e^{j n \frac{2 \pi}{N_0} k }$$ In these equations, $\omega_0 = 2 \pi / N_0$ is the fundamental frequency of the periodic signal in time and $N_0$ is the period of the signal.
When we compute the "Fourier Transform" on the computer, we are actually using the Discrete Fourier Transform (DFT). Since computers cannot hold continuous information, the time domain and frequency domain representations must both be discrete. Hence, the DFT is very similar to the Discrete-Time Fourier Series.
The core difference between the DFT and the Discrete Fourier Series is that the DFT assumes a signal always starts at $n=0$ and the fundamental period is always the length of the finite signal $N_s = N_0$. That is, the DFT sum goes from $n=0$ to $n=N_0-1$ and the fundamental angular frequency is $\Omega_0 = 2 \pi / N_0$. In addition, there is a slight normalization difference between the two transforms.
Therefore, the DFT is defined by $$ X[k] = \sum_{n = 0}^{N_0-1} x[n] e^{-j n \frac{2 \pi}{N_0} k} $$ $$ x[n] = \frac{1}{N_0} \sum_{k = 0}^{N_0-1} X[k] e^{j n \frac{2 \pi}{N_0} k} $$ This implies that to get discrete signals in time and frequency, we essentially assume the signals are periodic in both time and frequency (but only ever compute the result from one period).