Schoenberg's concept of 12-tone music means the application of 12 unique pitches in a specified sequence or "row". To make a 12-tone row he puts the 12 different notes of the chromatic scale in a desirable order without repeating any of them.
Ex. - G, F#, B, G#, E, A#, A, C, D#, C#, D, F
This row is also usually used in 3 different variations: upside down (inversion), backwards (retrograde), or upside-down AND backwards (retrograde inversion).
So that's all I knew for the last 20 years basically about "dodecaphonic music". In the last year, from conversations with a music educator friend and from the book above I've learned much more about 12-tone.
The 12-tone row doesn't have to be single notes going from left to right. It can also go horizontally, suddenly go backwards, then forwards, etc... The horizontal aspect really blew my mind. Theoretically the row could be written as just be one 12-note chord - or an 11-note chord followed by 1 note. Basically any horizontal or vertical direction is allowed and you can even "go around" notes and come back to them later at the end. The main thing is that the notes are not repeated.
But wait! Here's another eye-opener (at least for me). I just wrote that notes should not be repeated. That's not literally true. Each of the 12 notes in a row should be thought of as entities, not mere pitch frequencies. By entities, I mean each note in the row could appear as a trill or as a "tremolo". By tremolo I really mean that the note can be repeated - but you cannot have a different note between the reappearance of the same note (AAABBB is OK, AABBBA is not OK). Schoenberg uses the 12-note row more like a "concept" of 12 unique entities based on pitches. I suppose if one were to take that concept to an extreme, you could play a non-12-tone phrase, such as "Ode to Joy" in the key of A, then the key of A#, then B, etc....(or any order of keys) and you could say that was a 12-tone application to "Ode to Joy".
Finally, the row can be stated, and then immediately restated in any register. So you could have a row with the starting note F, then repeat the row in F#. The whole piece doesn't have to obey this same sequence over and over again. The row can be treated pretty much as a motif. Basically as long as you can draw a circle around a bunch of notes and none of them "repeat" (see above) then that's a 12-tone moment. The next phrase can be in any key really.
Finally - and this is probably the MOST IMPORTANT rule about 12-tone composition:
THERE ARE NO RULES.
What I mean is that 12-tone is a tool used to break free from traditional harmony of the 19th century and if at any time one decides that it would be nice to have that first note repeated at the end of the phrase - it's perfectly fine. The 12-tone system is more like a "recipe" kind of rule. It's a good way to get some new flavors but if you want to substitute lemon juice for vinegar it's OK, you can still call it salad dressing. Schoenberg himself broke the 12-tone rule all the time. Once, when someone wrote to him pointing out that the had repeated a note within his tone row, Schoenberg basically told him that his music was to be listened to, not studied.
"My works are twelve-tone compositions, not twelve-tone compositions."
(Schoenberg letter to annoying nit-picking fan)
I highly recommend the website of the Schoenberg Center. There you can listen to every single Schoenberg work for free.
A perfect example of the 12-tone technique at work is Op. 25, Suite for Piano, Nr. 5: Menuett. Trio (1923). This was one of the earliest and most succinct 12-tone works Schoenberg wrote. The Austin book analyzes the rows in that piece very well.