Step 1: Value. Not all yards are created equal, due to differing implications for the probability of picking up a first down. A four-yard gain is much more valuable on 3rd and 4 than it is on 3rd and 15. So DVOA begins by assigning a success value to each play based on the down/distance context in which it occurred.
Step 2: Over Average. As other elements of game context change, so does the importance of achieving plays that are nominally successful by the standard established in Step 1. When trailing by 17 points with 5 minutes left, picking up a first down is of little consequence. DVOA compares the success value of plays to the average value to be expected, based a database of plays in comparable game situations, in order to account for this.
These first two steps generate VOA, which can be a useful stat in its own right.
Step 3: Defense-adjusted. This is actually something of a misnomer: "opponent-adjusted" would be more accurate. Given the small number of games in an NFL season and the division-heavy schedule, some teams will find it much easier than others to achieve (context-adjusted) per-play success. So the final step is to adjust VOA in comparison with the baseline implied by the opponents faced.