In short, earthquake focal mechanism solutions are the beach balls that appear on maps like those below. They are produced by compiling and analyzing the waves that are produced by an earthquake and recorded by seismographs. As the handout explains, they provide a wealth of information, including information on the type of fault motion that occurred, that is was it a strike-slip, reverse, or normal fault.
Some of the compilations of historic focal mechanisms show exceptionally well the modern state of deformation in different regions of the world. For example, the most recent Peruvian earthquake occurred on a thrust fault (dipping at ~27° according to the USGS). Most of the beach balls shown from historic earthquakes are the same character.
Which are different from strike slip faults, which dominated in the two images below.
Learning to qualitatively read these focal mechanisms exponentially increases the amount of information you can read from earthquake reports. The USGS also has a good figure showing the basics of the beachball diagrams. It is by the analysis of these waves, by the way, that geologists have determined that the recent coal mine collapse in Utah was not caused by an earthquake (also here) but instead produced the earthquake sensed by regional seismographs.