Super Grover

Question: When Conan is learning how to use a sword, why did the teacher slap Conan in the face and then kick the other student in the chest?

Answer: Like an Army Drill Sargent, he was berating them for not doing it better. Toughening them to become the best they could be.

Also, the other student laughs at Conan's punishment, so the teacher kicks him to teach him some humility.

What exactly did Conan do wrong to make the teacher slap him?

This war master is reinforcing in Conan the "discipline of steel." The war master's gestures imply that he was not pleased with Conan's focus, his stance, and his sword/blade control. So the slaps were to further ingrain his rigorous training, in pursuance of Conan's mental and physical discipline.

Super Grover