Buddha-mind (Chinese foxing, Japanese busshin) refers to bodhicitta, "[the] Buddha's compassionate and enlightened mind," and/or to Buddha-nature, "the originally clear and pure mind inherent in all beings to which they must awaken."
Regarding awakening, Harold Stewart explains:
In Buddhist terminology this all-decisive moment is known as the Awakening of the Buddha-Mind, or Bodaishin [...] There are three practically synonymous terms in the Mahayana for this: Bodaishin (Sanskrit: Bodhicitta); Busshin, literally 'Buddha-Heart' of Great Compassion (Sanskrit: Tathagatagarbha, or the latent possibility of Buddhahood inherent in all beings); and Bussho (Sanskrit: Buddhata), or the Buddha-nature.
Busshin may also refer to Buddhakaya, the Buddha-body, "an embodiment of awakened activity."
Chan/Zen is also called foxin zong (Chinese) or busshin-shū (Japanese), the "Buddha-mind school."