The boy seems to be named Ikuto and he is a catboy. He throws his cat claws at Tadase and then Tadase dodges.
Ikuto's chibi flying thingy says that what Ikuto and him are doing is crap. They leave.
So, Tadase saved Amu and now Amu is bleeding. Amu goes home. The little chibi flying things are called guardian characters, chara for short.

The chara came because Amu wished to have a different personality and the chara helps her with that. The charas's name is Ran. At school the next day, Amu is praised for saying that she loved Tadase in public like that. Amu is invited to go to the royal garden, where all the guardians talk. The queen's chair, Nadeshiko, gives Amu a note from Tadase about the eggs.
Meanwhile, the second chara is born. In gym class, Amu does better than anyone. The 2nd egg is not there anymore. In art class, amu usually draws horrible but know she draws good. I don't know why because Ran isn't an art guardian, she is a sports guardian. It was the 2nd egg! Her name is Miki. Amu chases Miki down. Miki is about to vanish because Amu doesn't want them. Amu tries to believe in Ran and Miki and they don't vanish. Ikuto is thinking that somewhere the character egg has returned. That is what Ikuto wants. A girl named Utau wants to get the embryo for him, which you don't know about yet. It's after school and Amu comes to the royal garden.