Annabeth is Percy's friend and sometimes competitor. Her mom is Athena, so you can imagine that she is one smart cookie. Thanks to Annabeth, we learn pretty quickly that Athena and Poseidon don't get along very well. Athena once caught Poseidon and his girlfriend in her sacred temple, getting busy. Another time, Athena and Poseidon competed to be patrons of the city of Athens, and Athena won. Once, though, Athena and Poseidon collaborated together inventing the chariot – Athena building the chariot part and Poseidon creating horses from ocean waves.
Like her mom, Annabeth is really good at strategizing, and she nearly always has a plan. That's why she is captain of her Capture-the-Flag team – she knows how to anticipate what the other team will do and what their weaknesses are. She also loves to study architecture. When the trio visit the Gateway Arch in St. Louis, Annabeth is super-excited. She has learned everything about how the Arch is constructed and why it is special. Her mom, too, shares a passion for building, inventing, and crafting things.
Initially, Annabeth is introduced to Percy as "just a camper" who has been at Camp Half-Blood "longer than just about anybody" (5.60). Percy describes her this way:
She was probably my age, maybe a couple of inches taller, and a whole lot more athletic looking. With her deep tan and her curly blond hair, she was almost exactly what I thought a stereotypical California girl would look like, except her eyes ruined the image. They were startling gray, like storm clouds; pretty, but intimidating, too, as if she were analyzing the best way to take me down in a fight. (5.71)
Later, we learn that she has been at Camp Half-Blood since she was seven years old. At the age of seven, she ran away from home and linked up with Grover, Luke, and Thalia, making it all the way to Camp Half-Blood safely. She witnessed Thalia's death first-hand. Since then, she has stayed at camp year round.
Annabeth and Percy have something in common: both have family issues and terrible step-parents. Her dad is a human, a professor at West Point who teaches American History. Annabeth is angry at him for rejecting her at an early age:
"My dad's resented me from the day I was born, Percy," she said. "He never wanted a baby. When he got me, he asked Athena to take me back and raise me on Olympus because he was too busy with his work." (13.27)
Later, he married a mortal, and Annabeth's stepmom treated his daughter like a freak, never letting Annabeth play with her younger step-brothers. Monsters were always sniffing Annabeth out, and Annabeth got blamed at a young age for putting her family in danger. After a while, the little demi-goddess couldn't take it anymore and ran away for good. Can you imagine running away from home at the age of seven? She must have been truly miserable.
Two summers ago, Annabeth's dad wrote her a letter telling her that he loved her and asking her to please come home and live with him full-time. He included his college ring, which she now wears around her neck. Annabeth took him up on his offer, but only stayed with him for three months. Her stepmom was terrible as always, making Annabeth feel like a constant outsider in the family. Annabeth soon returned to Camp Half-Blood as a full-time camper.
Over the course of their quest, Percy encourages Annabeth to keep in touch with her dad. He says, "you shouldn't give up […] you should write him a letter or something" (16.93). At the end of The Lightning Thief, we learn that Annabeth has taken Percy's advice and has decided to have another go at living with her dad for the school year.
Annabeth wants a challenge. She wants a quest more than anything. On a quest, a demi-god's powers are truly tested, and all of the work and training that she or he has done at Camp Half-Blood is put to the test. After they battle the Furies on the Greyhound bus, Annabeth chastises Percy for jumping into the fight. She tells him, "it's just that if you died…aside from the fact that it would really suck for you, it would mean the quest was over. This may be my only chance to see the real world" (11.15). Annabeth has been cooped up at Camp Half-Blood longer than just about any other camper. It must get kind of boring and kind of suffocating for her. She wants to have an adventure and gain some life experiences. We totally understand that.