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While studying photometric redshifts, I came across the ugriz (or u'g'r'i'z) system for classifying magnitudes of galaxies, but I didn't find much information on the internet about how it works.
Can somebody explain that? Do the letters "ugriz" have any special meaning, or is its just a nomenclature?
"ugriz" is short for U-band, G-band, R-band, I-band, and Z-band, detailed on the Wikipedia article on photometric systems: $$\begin{array}{|c|c|} \hline \text{Band}&\text{Effective Wavelength Midpoint }(\lambda_{\text{eff}})\\ \hline \text{U} & \text{365 nm}\\ \hline \text{G} & \sim\text{475 nm}\\ \hline \text{R} & \text{658 nm}\\ \hline \text{I} & \text{806 nm}\\ \hline \text{Z} & \text{900 nm}\\ \hline \end{array}$$ This $ugriz$ system is slightly different from the $u'g'r'i'z'$ system; conversions can be found here. The starting point is the same (note that $u(\text{2.5m})=u'$), but there are slight vertical and horizontal shifts.
The differences between $ugriz$ and $u'g'r'i'z'$ are much smaller than the differences between $u'g'r'i'z'$ and the $UBV(RI)_C$ system, which was replaced by them both. This older system of filters had large overlaps and did not cover as broad a range. See Fig. 1 from this thesis for a comparison:
