Bhajan is the generic name for any kind of Indian, usually Hindu, devotional song. It is completely text-led, its devotional nature underpinned by the words rather than by any specific musical style. It can be something as straightforward as a recitation or chant (kirtan) of a given mantra (usually a word or sound repeated to aid concentration in meditation) all the way to something as complex and sophisticated as the Dhrupad of North India or the kriti form of Carnatic music, based on pure raag (melodic structure) and executed in a specific taal (rhythmic cycle).
The word bhajan is derived from the word bhakti meaning 'loving devotion,' also the name of a specific spiritual movement which originated in South India and rapidly spread North and from which a tremendously enduring genre of mystical poetry emerged. Followers of the bhakti movement were opposed to the dogmatic nature of religious ritual and maintained that God was omnipresent, without requiring shape or form.