Avatar means "one who descends." By this is meant one who is established in Krishna Consciousness or Brahman Consciousness and purposefully comes into our world to help us rise to Brahman Consciousness.
Avatars are of two types - Krishna Avatar and Jîva Avatar. Avatars that arise directly from Krishna are very rare. The Supreme Dieties of our universe are Krishna Avatars: Narâyana, Vishnu, Rudra and Ganesha. Lord Brahma is the foremost Jîva Avatar and all those who are in Brahman Consciousness are Jîva Avatars.
The mission of all Avatars is to help jîvas advance in their course of evolution back to Unity with S'iva.
Krishna Avatars who incarnated on earth include Parashuram, Lord Buddha, S'rî Krishna, S'rî Balarama, and S'rî Ram. There have been many Krishna Avatars that did not take human physical form, such as Matsya, Kurma, Varaha, Narasingh, and Vaman.
The Vedic tradition indicates there is another Krishna Avatar to come at the end of Kali Yug - Lord Kalki. Kalki will replace all the leaders of society with caring Jîva Avatars to establish the immediate rise of a Sat-yuga.
Some incorrectly believe that Kalki will be the one to establish the 10,000 year Golden Age that is due to begin now. That is not correct. There will be Jîva Avatar incarnations of Krishna who will carry out this task. The most notable Jîva Avatar, working on this task in the tradtion of Sri Ram, is Maharishi Mahesh Yogi who has for the past 50 years been striving to create this 10,000 year Golden Age or "Heaven on Earth" through His Transcendental Meditation technique, World Government for the Age of Enlightenment, and other programs for the development of consciousness. A few years ago He inspired the organization of the Natural Law Party in many countries to provide enlightened political leadership. Another Jîva Avatar, who will be identified with Bhagavan Sri Krishna, will arise soon to destroy the power of the corrupt world leaders as the final stroke that will bring in the Golden Age.
Recently, S'rî Vyuha was organized by Vis'vamitra, who is directly inspired by Maharishi, to provide entry to the path to Brahman Consciousness for everyone.