What is an avatar?
It is a soul who has transcended all past karma, all identification with anything other than God alone, and is fully liberated in God, but chooses to return to earth to help others.
In the book, "Conversations with Yogananda" (conversation #87), Swami Kriyananda quotes Paramhansa Yogananda as saying that an avatar is "a full manifestation of God," with "divine power to shower blessings, generally, on all mankind.
Those who still have some past karma of their own to work out, return primarily to help their own disciples. These saints, called jivan muktas ("freed while living"), are able to uplift a few, but cannot carry innumerable disciples to God.
Those, on the other hand, who return without any karma of their own, having become param muktas ("fully liberated") in a former life, come as avatars. Whoever comes to them for help can be saved, though his salvation is not necessarily immediate."
On another occasion, Swami Kriyananda asked Yoganandaji whether he (Yoganandaji) was an avatar. “It would take such a one,” he replied quietly, “to bring a mission of this importance.”