When you attune to something, you adjust to it and become aware of the way it works. A new parent has to attune to a baby's schedule and personality.
Preschool teachers need to attune to their students' various needs and interests, and when you visit a foreign country it's important to attune to the unfamiliar culture, so that you're both comfortable and respectful. The verb attune is almost always followed by "to," and it often shows up in the form of "be attuned to." Attune comes from tune, "bring into a state of proper pitch."