Charisma in the study of religion is a technical term from Max Weber, who took the Greek word for a gift of grace, used by Paul for the gifts of the Spirit, and made it one of his three ideal types of legitimate authority.
Weber distinguished traditional authority, resting on custom and inherited status; legal-rational authority, resting on impersonal rules and offices; and charismatic authority, resting on a particular person's perceived extraordinary qualities and on the devotion those qualities inspire. The crucial point of the analysis is that charisma is a relationship rather than a property. It exists only so long as followers recognise it, and it must be continually validated by success, which makes it inherently unstable, a prophet whose predictions fail loses it.
That instability produces the concept's most useful application, the routinisation of charisma. A movement founded on personal authority faces a crisis when the founder dies, and it must convert charisma into something transmissible or dissolve. The available routes are few and they recur: designation of a successor, hereditary succession, election by a body, an office to which charisma attaches independently of its holder, or a text. Weber's analysis maps directly onto actual religious history. The Sunni-Shia division over the succession to Muhammad is at bottom a dispute over which route is legitimate. The development of episcopacy and apostolic succession converts personal authority into office. The Sikh vesting of guruship in scripture and community is an explicit and unusually deliberate resolution of the problem. And the histories of new religious movements after a founder's death follow the same limited set of paths with striking regularity.
The concept has limits and has been criticised. It can be applied circularly, since charisma is identified by the devotion it produces and then used to explain that devotion. It concentrates attention on the leader rather than on the social conditions that make a following possible, and later scholarship has emphasised that charisma is attributed rather than possessed and requires a receptive audience with its own needs. And in ordinary usage the word has drifted to mean personal magnetism, which is much weaker and more individual than Weber's sense.
The original theological sense remains in use and should not be confused with the sociological one. In Christian usage the charismata are the gifts of the Spirit listed in 1 Corinthians 12 and Romans 12, and the Charismatic movement takes its name from them, being concerned with prophecy, healing and tongues rather than with the authority of leaders, though in practice the two senses meet more often than either would prefer.