The saktyavesa-avataras are of two types: forms of divine absorption (bhagavad-avesa), and divinely empowered forms (saktyavesa). All saktyavesa-avataras are jiva-tattvas, including those who are considered to be incarnations. When a jiva-tattva becomes specifically empowered by the Lord, he is called saktyavesa avatara. Dhanvantari, for example, who was the inaugurator of medical science, was a jiva-tattva saktyavesa incarnation of the Supreme Personality of Godhead.
The primary saktyavesa-avataras are directly empowered by the Supreme Personality of Godhead and are called an incarnation. Lord Brahma and Lord Siva are considered to be such saktyavesa's, as are Lord Buddha, Kapiladev, Rsabhadev and Nara-Narayana. The secondary saktyavesa-avataras are indirectly empowered by the Supreme Personality of Godhead and are called vibhuti. Jesus Christ and Hazrat Mohammed are included in this group, as are the great Vaisnava Acaryas like Srila Bhaktivinoda Thakura, Srila Bhaktisiddhanta Saraswati Thakur, and A.C. Bhaktivedanta Swami Srila Prabhupada.
"Generally the eternal associates of the Lord in the spiritual sky do not come to the material world. Sometimes they come, however, by the order of the Lord--not to hold any administrative post, but to associate with the Lord in person or to propagate the message of God in human society. Such empowered representatives are called saktyavesa-avataras, or incarnations invested with transcendental power of attorney."
Srimad-Bhagavatam 3:5:21 Purport
"The Lord then pointed out that there is no limit to the saktyavesa-avataras and that they cannot be counted. However, some can be mentioned as examples. The saktyavesa incarnations are of two kinds--direct and indirect. When the Lord Himself comes, He is called saksat, or a direct saktyavesa-avatara, and when He empowers some living entity to represent Him that living entity is called an indirect or avesa incarnation. Examples of indirect avataras are the four Kumaras, Narada, Prthu and Parasurama. These are actually living entities, but there is specific power given to them by the Supreme Personality of Godhead. When a specific opulence of the Supreme Lord is invested in specific entities, they are called avesa-avataras. The four Kumaras specifically represent the Supreme Lord's opulence of knowledge. Narada represents the devotional service of the Supreme Lord. Devotional service is also represented by Lord Caitanya, who is considered to be the full representation of devotional service. In Brahma the opulence of creative power is invested, and in King Prthu the power for maintaining the living entities is invested. Similarly, in Parasurama the power for killing evil elements is invested. As far as vibhati, or the special favor of the Supreme Personality of Godhead, is concerned, it is described in the Tenth Chapter of Bhagavad-gita that a living entity who appears to be especially powerful or beautiful should be known to be especially favored by the Supreme Lord.
Examples of direct or saksad-avataras are the Sesa incarnation and the Ananta incarnation. In Ananta the power for sustaining all planets is invested, and in the Sesa incarnation the power for serving the Supreme Lord is invested."
Teachings of Lord Caitanya, Chapter 8
"There are unlimited saktyavesa-avataras of Lord Krsna. Let Me describe the chief among them. Empowered incarnations are of two types--primary and secondary. The primary one is directly empowered by the Supreme Personality of Godhead and is called an incarnation. The secondary one is indirectly empowered by the Supreme Personality of Godhead and is called vibhuti. Some saktyavesa-avataras are the four Kumaras, Narada, Maharaja Prthu and Parasurama. When a living being is empowered to act as Lord Brahma, he is also considered a saktyavesa-avatara. Lord Sesa in the spiritual world of Vaikuntha and, in the material world, Lord Ananta, who carries innumerable planets on His hood, are two primary empowered incarnations. There is no need to count the others, for they are unlimited. The power of knowledge was invested in the four Kumaras, and the power of devotional service was invested in Narada. The power of creation was invested in Lord Brahma, and the power to carry innumerable planets was invested in Lord Ananta. The Supreme Personality of Godhead invested the power of personal service to Lord Sesa, and He invested the power to rule the earth in King Prthu. Lord Parasurama received the power to kill rogues and miscreants. Whenever the Lord is present in someone by portions of His various potencies, the living entity representing the Lord is called saktyavesa-avatara--that is, an incarnation invested with special power."
Caitanya-caritamrta, Madyam lila 20:367-373
Bhaktivedanta Book Trust. HDG A.C. Bhaktivedanta Swami Srila Prabhupada.