A-440Hz is middle A, so 2x that frequency or 880Hz. is one octave up.
Half an octave is for this frequency 440 + 220Hz or 660 Hz.
Similarly 1/3 is dividing that octave into thirds.
2/3rd of an octave is 2x 1/3 of that octave above some frequency.
This usually refers to smoothing or the bandwidth of an EQ adjustment.
In the case of smoothing, all the energy on either side of a center frequency (decided in the hardware or software) is averaged and that output is what is shown or adjusted (respectively for measurement or smoothing). For measurement this yields a smoother and flatter looking response.
Fully unsmoothed response curves have lots and lots of ups and downs, so this makes it easy to see the trend... of course overly smoothed curves make things look better than they are and could possibly ignore some actual response anomaly.
Balanced means that there is a dual polarity signal plus ground. Three wires. Unbalanced means a single polarity signal and ground.
Typically the balanced connector is an XLR or a TRS phone plug (3 wire or "stereo" 1/4" phone plug). The unbalanced is typically an RCA plug or else a mono 1/4" phone plug (tip and sleeve - two wires).
Balanced is associated more with pro gear. The balanced line gives better immunity from hum and noise, especially on longer runs, and if true professional design is "600 ohm" meaning it runs lower Z which helps prevent HF rolloff - again better on longer runs.