Magnitudes are a measurement of an objects brightness, for example a
star that is 6th magnitude is brighter than a star that is 11th magnitude.
The term arcsecond comes from an arc being divided up into seconds. There
are 360 degrees in an circle, and each degree is divided into 60 minutes,
and each minute is divided into 60 seconds. A square arc second has an
angular area of one second by one second.
The term magnitudes per square arc second means that the brightness in
magnitudes is spread out over an square arcsecond of the sky. If the SQM
provides a reading of 20.00, that would be like saying that a light of a
20th magnitude star brightness was spread over one square arcsecond of the
sky.
Quite often astronomers will refer to a sky being a "6th magnitude sky",
in that case you can see 6th magnitude stars and nothing dimmer like 11th
magnitude stars. The term "6th magnitude skies" is very subjective to a
persons ability to see in the night, for example I might say "6th
magnitude skies" but a young child with better night vision might say "7th
magnitude skies". You can use this nifty calculator created by SQM user K. Fisher to do that conversion,
or this chart.
The "magnitudes per square arcsecond" numbers are commonly used in
astronomy to measure sky brightness, below is a link to such a comparison.
See the third table in section 10 for a good chart showing how these
numbers in magnitudes per square arcsecond relate to natural situations:
www.stjarnhimlen.se/comp/radfaq.html
Each magnitude lower (numerically) means just over 2.5 times as much
more light is coming from a given patch of sky. A change of 5 mags/sq
arcsec means the sky is 100x brighter.
Also, a reading of greater than 22.0 is unlikely to be recorded and the
darkest we've personally experienced is 21.80.