There are quite a few struggles for power in Rick Riordan's The Lightning Thief, from Titan to Olympian god to demigod. Kronos — the Titan father of the elder Olympian gods — wants to overthrow his children. Ares — the god of war — tries to start a war between Zeus, Hades, and Poseidon. Then there's Luke — a demigod who feels mistreated by the gods and those in charge at Camp Half Blood.
We learn early in the story that Kronos was defeated by his children and sent to the Underworld after devouring and then regurgitating the gods:
Zeus did indeed feed Kronos a mixture of mustard and wine, which made him disgorge his other five children, who, of course, being immortal gods, had been living and growing up completely undigested in the Titan's stomach. The gods defeated their father, sliced him to pieces with his own scythe, and scattered his remains in Tartarus, the darkest part of the Underworld.
This is the backstory for the main power struggle throughout the book. It is revealed at the end that Kronos has been manipulating both the gods and demigods so that he can escape Tartarus, create chaos among the Olympian gods, and regain his throne.
Ares, whose power comes from his mastery of war, is among those manipulated by Kronos. After catching the real thief, he uses the stolen lightning bolt and helm of darkness to create animosity among the three elder gods (Zeus, Hades, and Poseidon), placing Percy Jackson in a position to take the fall for the theft. Though it was Luke who initially stole the Olympian gods' items, Ares is the one who sets everything up. When Percy confronts Ares towards the end of the book, the god of war spills the beans:
"The point is, kid, you're impeding the war effort. See, you've got to die in the Underworld. Then old Seaweed will be mad at Hades for killing you. Corpse Breath will have Zeus's master bolt, so Zeus'll be mad at him. And Hades is still looking for this..."
As you can see, Ares set in motion a series of events that were meant to lead to the three elder gods warring with each other. One thing to keep in mind, however, is that Ares was manipulated by Kronos, so this power struggle cannot be fully attributed to the god of war.
Finally, we get to Luke. While his is not a struggle for power per se, he does facilitate both Kronos' and Ares' struggles by stealing the lightning bolt and helm of darkness in the first place. Luke's struggle comes from his dissatisfaction at being neglected by his father, Camp Half Blood, and the rest of the Olympian gods. He does not seek power in the traditional sense. Demigods' power (i.e. glory) comes from their ability to take part in quests. Luke feels misused, particularly when it comes to the only quest:
"He showed me that my talents are being wasted. You know what my quest was two years ago, Percy? My father, Hermes, wanted me to steal a golden apple from the Garden of the Hesperides and return it to Olympus. After all the training I'd done, that was the best he could think up."
The irony here is that this quest is both a foreshadowing and a 'training' of sorts for his actions throughout the book. He is upset that he is asked to steal an item rather than engage in heroic battle, yet that is exactly what he does for Kronos.
Clearly, the main struggle for power comes from Kronos. The other two examples show a different kind of 'power' being sought, but they are done so through the Titans' efforts. Though his plans are foiled by Percy Jackson and his friends, these events set Kronos up as a force to be reckoned with throughout the other books in the series.