Well, the twelve tone system is composition based on the twelve cromatic notes, put into a particular order(it can be random) to make up a tone row.
This tone row can be altered in several ways and you can switch through these rows throughout your composition. Here are the different forms of the tone row:
Tone Row - Original Form
Retrograde - Backwards
Inversion - Up side down - taking the intervals of the original row and going up that interval instead of down and vice versa.
Retrograde Inversion - Upside down and Backwards
Tranposed - whole tone transposed up or down an interval
This Tone Row is divided into four three note "cells". Melodies and motives are formed in individual cells e.g. F#, G, D# and after using the notes move on to the next cell and use those notes, not returning to the cell until all the notes in the tone row are used. The twelve tone system place equal emphasis on all notes of the cromatic scale
Harmony is made in these individual cells and doesn't form many consonant harmonies, the tritone is a common harmony in twelve tone composition.
Remember that twelve tone compositions are completely atonal and you can try to develop motives through rhythms and dynamics.
These are reasonably specific rules, but its pretty flexible. Hope that helps a bit.