An Alpha is the leader of a Shape-shifter pack and its highest ranking member.
The Alpha is the leader of the wolves and supposedly in control of the pack and responsible for it. After he rises to Alpha status, Jacob mentions feeling paternal towards the other wolves in his pack when there is a threat to it, as if he should be between them and the danger.
In the original La Push pack, Ephraim Black was the Alpha. Sam Uley became the next one, since he was the first to phase. Jacob Black was offered the leadership when he also phased, but refused. Jacob later became the Alpha of his own pack after an argument between him and Sam. Traditionally, the Alpha of the pack is the leader of the entire Quileute tribe.
The position of Alpha could also apparently be fought over, but it seems the lineage of the wolf has more to do with it. When Jacob took his rightful place as Alpha to keep Sam from attacking the Cullens, Sam thought that he meant to usurp him.
Jacob explains his fight for wolf claim as "My body was responding to the challenge. The primitive core of my wolf sense tensed for battle of supremacy."
Like all werewolves, Alphas are able to communicate telepathically to their pack-mates. Indeed, all thoughts are shared, whether the individuals like it or not. They can also talk to other Alphas from different packs, though this communication is far from indiscriminate. Their power comes from their pack, and they hold a supernatural control over it; if they command something to happen, the wolves must comply whether they want it or not.
In addition, should the Alpha's mind become magically protected from harm, the other wolves' will also be protected. This was proved when Bella Swan shielded Sam and Jacob with her power, and provided protection to the entire packs. This may also mean that if the Alpha falls victim to a supernatural psychic power, the rest of the pack will go through the same effect and become open targets.