I watched Breaking Dawn Part One, where they used the term Alpha Wolf for their leader. But how can one become an Alpha wolf in the Twilight series, and what are the differences between the Alpha and Beta wolves? It doesn't get explained or maybe I missed something. Is this explained in Breaking Dawn Part Two or the novels etc?
In Twilight, the leader of the tribe is the Alpha Wolf.
The second in command is referred to (by Bella, not the shifters themselves) as the Beta Wolf.
How can become a Alpha wolf?
Either by election, or by lineage.
The main power an Alpha gets is the "Alpha Voice".
The Alpha voice is an ability that only a genetic or elected Alpha shape-shifter possesses. It is unknown how the Alpha voice works and whether an Alpha shape-shifter can use his Alpha voice to command his pack members in both human and wolf forms. Once an Alpha shape-shifter gives orders to members of his pack, his orders are obeyed blindly. The pack members cannot directly disobey commands even if they are in disagreement. This ability, however, is limited to pack member and cannot influence another Alpha shape-shifter.
The Beta Wolf can also do this, but can be overruled by the Alpha.
In addition an Alpha can also communicate with the Alphas of another pack telepapthically (like how they can within their own pack).