The logical relation of numerical sameness, in which each thing stands only to itself. Although everything is what it is and not anything else, philosophers try to formulate more precisely the criteria by means of which we may be sure that one and the same thing is cognized under two different descriptions or at two distinct times. Leibniz held that numerical identity is equivalent to indiscernibility or sameness of all the features each thing has. But Locke maintained that judgments of identity are invariably made by reference to types or sorts of things. The identity of individual persons is an especially troublesome case.
(Via Philosophical Dictionary)
Philosophers of mind who hold to naturalism typically assert some kind of identity relation between the mind and the brain, which many theists find problematic. J. P. Moreland has done excellent work in defending a substance dualist view of persons that grounds individual identity in the human soul rather than any biological properties.