A guardian is someone charged with protecting a summoner throughout a pilgrimage across Spira to defeat Sin. Tidus' father Jecht was one of Yuna's father's guardians. Tidus became one of Yuna's.
With six guardians, Yuna had more than most summoners. She stated that each of her guardians was someone she could trust with her life, though, and that was a huge compliment to Tidus, who had just entered her life.
It's hard to say what made Yuna ask Tidus to be her guardian since she didn't know him that well. I think it was the fact that Tidus was Jecht's son, and Jecht was someone she trusted and admired. I think she also bonded with Tidus, and formed an easy connection. I think she knew he was someone who could be trusted.
Tidus was proud to be a guardian, and referred to himself as one time and time again. When Rikku was worried about Yuna sacrificing herself to defeat Sin, and when Lulu and Auron were concerned about Yuna being kidnapped, Tidus spoke up and said they would protect her; they were her guardians.
Tidus had his own view of what a guardian was, and used it to benefit Yuna. When Kimahri was holding off Seymour so the party could escape, Yuna wanted to go back and help Kimahri, and Auron was telling her to leave. He said, "He is a guardian. Protecting you is everything." Tidus used Auron's quote to get Yuna what she wanted.
That's right! We're all guardians! Yeah, and you know what that means? Yuna... Anywhere you go, I'll follow!
Tidus taught Yuna to whistle, and said should they ever get separated, whistle, and he'll come running. As Yuna's guardian, Tidus would follow her to the end of the world if he had to. Being a guardian meant more to Tidus than simply protecting Yuna. It meant being with her, and being there for her.
Yuna: Stay with me until the end. Please.
Tidus: Not until the end... Always.
When Tidus learns that summoners die at the end of their pilgrimage summoning the Final Aeon, Tidus dedicates himself to finding a way to save Yuna.
I won't let you die. I'll find a way, somehow.
Like everything in Yevon, there is more than what meets the eye in the role of a guardian. When the party meets Yunalesca to obtain the Final Aeon, they learn that a summoner is not alone in giving up her life; a guardian must give up his or her life as well, to merge with Yunalesca's aeon and become the Final Aeon.