What is the ugriz magnitude system? 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:
$$\begin{array}{|c|c|}
\hline \text{Band}&amp;\text{Effective Wavelength Midpoint }(\lambda_{\text{eff}})\\
\hline \text{U} &amp; \text{365 nm}\\
\hline \text{G} &amp; \sim\text{475 nm}\\
\hline \text{R} &amp; \text{658 nm}\\
\hline \text{I} &amp; \text{806 nm}\\
\hline \text{Z} &amp; \text{900 nm}\\
\hline
\end{array}$$
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: