The term "serialism" is sometimes used synonymously with "twelve tone music". The truth is, however, that the 12-tone method is just one of several kinds of serialism. The reasoning behind total serialism is simple enough: if Schoenberg could compose notes according to numerical patterns (or serialize them), then why couldn't you do the same thing with other aspects of music? Going down this train of thought, composers quickly came up with ways to serialize all kinds of musical elements: note length, silence, texture, volume, and so on.
This could be done in several different ways. A serial composer could have several different series to govern several different elements of the music (for example, one series for the notes, one for the note lengths, another for volume, etc...). Another way of doing things would be to have everything be derived in one way or another from a single numerical series. Either way, the composer would be close to having "total control" over every little detail of his piece by way of the series he came up with.
Of course, this doesn't mean that composers could just pick a few random numbers and try to turn them into music. The real challenge to writing "good" serial music is to somehow arrange the series so that the resulting music at least makes some degree of sense; otherwise, the whole thing can easily wind up sounding like total randomness (see indeterminism if you want total randomness.) Even in the most carefully constructed serial works, however, the unaccustomed listener will probably get the impression of randomness.
Serial music is almost completely detatched from "traditional" music, in terms of melody, rhythm, and harmony. Ever since the days of Bach or Haydn, music had been written so that the listener could follow its development from beginning to end. This was done through "logical" harmonic progressions, melodic lines, and a sense of pulse or rhythm. In serial music, the listener is only aware of unrepeated and unpredictable musical "events" which dissolve in and out of each other in an apparently random fashion. The end result: it's usually very complex and is usually understood only after many listenings (if ever).
Total serialism followed the success of the 12-tone series, and developed around the
beginning of the 1950's. It continued into the 1960's with such
composers as Igor Stravinsky, Milton Babbit, Pierre Boulez, and