Important Characters with regards to the Ideals of Shirou
Saber is a genderbent version of King Arthur. She is summoned by Shirou by accident to avoid getting killed by Lancer, and in each route, she starts with her ideal of changing how her reign ended. However, in each route, this ideal is either thrown away or completely corrupted, the former happening in Fate and Unlimited Blade Works, and the latter occurring in Heaven's Feel(the latter is confirmed by supplemental material) through her interactions with Shirou, Archer, and Sakura.
Shirou is an overly idealistic teenager whose sole desire is to become a hero. As a child, the previous Grail War was held, burning down the section of the city that he resided in. He remembers nothing of his life before the fire, and, after being adopted by his father, takes on the ideals that his father has thrown away: to become a hero of justice. It is his sole purpose for existing, exemplified by the statement "I don't want to become one... I will become one"! His reckless idealism only gets him into more and more trouble as the story continues, particularly with the Servant Archer in the Unlimited Blade Works route.
Each servant is summoned into it's own class, and the Archer class servant is one that resonates with Shirou. He is the personification of Shirou's ideals, a version of Shirou from the future, who has lost everything becoming a hero. Even after he dies, hanged by those that he saved, he still acts as a hero, forming a contract with an entity known as the Counter Force, in order to continue acting as a hero after his death. The Counter Force, or Alaya is Humanity's collective subconscious, interested in self-preservation above all else, and it's agents constantly enforce Alaya's will to survive, killing any threat to humanity's collective existence, especially humans themselves. Alaya and her guardians exist outside of time, and thus, Archer is forced to kill people over all of human history -- breaking him into a bitter, cynical man. He whose dream was to act as a hero and see people smile is forced to bear witness to thousands of tears and expressions of agony, meaning that, essentially, his ideals betrayed him in the cruelest possible fashion. Thus, once Archer is summoned into the war by another master, he bides his time, watching Shirou's behavior and sees if Shirou will end up like Archer did.