For rare systems, like Algol, the period is observable to the naked-eye. Algol was seen to brighten and dim over the course of almost three days. For most variables, the fluctuations are too dim to be seen by eye. Moreover, there is a huge range of variability timescales. Some variables fluctuate on timescales of a few seconds (e.g., X-ray bursters), while others may take several years (eruptive T Tauri stars).
It's important to know your period range of interest, which is dependent upon two things (a) the integration time --- you cannot search for periods less than the integration time and (b) the total observation window.\[P_{\left\{\max\right\}}\le\frac{1}{3}\tau\]
In signal processing, a periodogram is an estimate of the spectral density of a signal. Periodograms are super handy in determining the optimal period of a time series. A periodogram is similar to the Fourier Transform but is optimized for unevenly time-sampled data, and for different shapes in periodic signals.
They come in many flavors and it is important to know the strengths and weaknesses of those that you use. So far we have employed three basic types:
  1. Box Least Square Periodogram
  2. Generalized Lomb Scargle Periodogram
  3. Phase Dispersion Method Periodogram
Let's continue our focus on the overall picture and avoid getting into the weeds. We will come back to how these periodograms work later. 
How do you know what you're looking at?
Some characteristics to look for: