An Alpha is a title given to the dominant male or female leader of a group of animals, or it can be achieved by challenging and defeating the current alpha.
Rise of the Planet of the Apes
Caesar's father was the alpha of his group.
Rocket was the alpha of the group at San Bruno Primate Shelter, establishing his dominance over Caesar by ripping his shirt off and attacking him. His wife Tinker shared this role as his wife. Caesar later asserted his status as the alpha male by surprising Rocket, hitting the ape down and had Buck force the ape to submit leadership to him.
Caesar then went on to lead the apes in their escape to the red woods, orchestrating their escape and combat across the bridge. He also established his dominance over the new member Koba, shoving him to the ground when he tried to harm Will Rodman. He started the phrase "Apes, together, strong" that the group confidently follow.
Dawn of the Planet of the Apes Comic Books
Pope led the army two years later, leading them on training missions, scouting missions into human houses and the San Francisco Zoo and giving them orders against Caesar's beliefs. When a large tribe of primal apes was discovered in the zoo, Pope challenged the alpha of the tribe to battle and killed him (albeit by cheating), making him the alpha over both groups of apes. He led his evolved apes through ruthless intimidation while he led the primal apes through fear and dominance. He also beat and killed apes who defied him.
After Koba killed Pope he became the new alpha of the primal apes and led them back to Caesar's Colony and they eventually followed Caesar too.
Dawn of the Planet of the Apes
Caesar still led the group ten years on, leading them in charges against deer in the wood, was the head of the Ape council and did the majority of the talking with humans. He has the say over what to do when Malcolm came with his group. He also further established dominance by countering Koba's constant challenges, making the bonobo submit by holding out his hand as a sign of respect. He also later beat him to near death for his callous accusations.
Koba eventually shot Caesar and established himself as the new alpha, leading the Ape army in the attack on the New York settlement, inspiring his troops by charging singlehandedly into the fire on a horse and also secured them a tank to use. He led in fear rather than respect, killing apes (like Ash) and imprisoning others who defied him.
Caesar later returned to challenge him, successfully defeating the bonobo and regained leadership.
The Last Frontier
Khan led his own tribe in a mountain area. He was respected by all apes, although his sons held some animosity. Upon Khan's death after a fire fight with a human group, he made the eldest son Tola the new alpha.
Tola initially led through aggression and inducing fear in the tribe although his younger brother Bryn (player determined) and Brutus remained loyal. However, following a severe defeat to the humans, Bryn lost faith in Tola and declared himself the new alpha. The pair fought intensely, with Bryn ultimately winning the fight by either choking Tola to death or their youngest brother Juno impaling Tola's abdomen, leading him to bleed to death. (player determined) However, if Bryn shows mercy and Juno is dead, then Tola recovers, beating Bryn to death with a rock retaining his leadership.
Bryn/Tola (player determined) then led the clan in a final assault against the human army, defeating them and securing the tribes survival in the harsh winter. Alternately, they fail, deciding to flee the area, Tola/Bryn (player determined) uncertain can lead them to safety through the dense snowstorm or to their demise if they attack the town.
War for the Planet of the Apes
Caesar still leads the apes 12 years on although his followers have become fractured. Several have defected from his group (notably Red) due to them being loyal followers of Koba, although the alpha personally speculates it is due to their fear of what he would do to them something Red coldly defies. The apes who are loyal follow him devotedly, particularly Rocket, Maurice and Luca, who stay with him during the apes personal vendetta against the Colonel. Caesar later inspires his group again when he defies Red's torture of one ape, and leads them in the planning to escape imprisonment.