Aatma means 'eternal self'. The aatma refers to the real self beyond ego or false self. It is often referred to as 'spirit' or 'soul' and indicates our true self or essence which underlies our existence.
There are many interesting perspectives on the self in Hinduism ranging from the self as eternal servant of God to the self as being identified with God. The understanding of the self as eternal supports the idea of reincarnation in that the same eternal being can inhabit temporary bodies.
The idea of aatma entails the idea of the self as a spiritual rather than material being and thus there is a strong dimension of Hinduism which emphasises detachment from the material world and promotes practices such as asceticism. Thus it could be said that in this world, a spiritual being, the atman, has a human experience rather than a human being having a spiritual experience.