U.S. Geological Survey image
The focal mechanism of an earthquake sums up (1) the slip motion of the rocks underground and (2) the orientation of the fault that the slip occurred on. (See Fault Types in a Nutshell for a quick explanation of the three basic fault types.)
The "beachball" diagram is built from seismograms recorded around an earthquake. The first motion of the arriving P-waves is checked for each seismograph station, to see if it's a pushing (upward/away from the source) or a pulling (downward/toward the source) motion. Pushing is compression; pulling is tension. This first-motion information is plotted as it would appear on a hollow half-sphere sunk in the ground, showing what directionboth compass direction and vertical anglethe seismic waves left the quake focus to reach each station. (The half-sphere is represented on flat paper as an equal-area projection.) Compressions are marked black, tensions are marked white, that is, as circles. In a well-behaved earthquake with well-recorded data, all the compressions cluster in two opposite quadrants. Same with the tensionsthey occupy the other two quadrants.
The fault that created the compression and tension runs exactly between black and white, where tension and compression balance. That's what the top diagrams (A) show, with the compressive areas shown black and tension areas shown white. The motion of the rocks at the source of the earthquake goes from white toward the black. (The P and T axes, shown by dots marking one end, are part of the moment tensor, which is the ultimate boiled-down mathematical description of the quake motions. You can ignore them.)
The trouble is that there are two solutions to the diagram, two planes along which earthquake motions could yield the same seismographic results. One is the real fault plane, therefore the other, called the auxiliary plane, has no physical meaning. Seismographic evidence alone isn't enough to choose the right one. That's why in (B) there are two different slip diagrams shown for each focal mechanism. Usually there are enough clues in the local geology or aftershock patterns to pick the right solution.
They teach this material to geology majors, but not every student retains it. I'm one of those guys that has to consult this diagram several times a year. For a much more thorough treatment of the subject including exercises, see this PDF by Vince Cronin of Baylor University, "A Draft Primer on Focal Mechanism Solutions for Geologists."