From the Bhāgavatam we understand that Lord Buddha is the incarnation of Kṛṣṇa who appeared when materialism was rampant and materialists were using the pretext of the authority of the Vedas. Although there are certain restrictive rules and regulations regarding animal sacrifice for particular purposes in the Vedas, people of demonic tendency still took to animal sacrifice without reference to the Vedic principles. Lord Buddha appeared to stop this nonsense and to establish the Vedic principles of nonviolence. Therefore each and every avatāra, or incarnation of the Lord, has a particular mission, and they are all described in the revealed scriptures. No one should be accepted as an avatāra unless he is referred to by scriptures. It is not a fact that the Lord appears only on Indian soil. He can manifest Himself anywhere and everywhere, and whenever He desires to appear. In each and every incarnation, He speaks as much about religion as can be understood by the particular people under their particular circumstances. But the mission is the same—to lead people to God consciousness and obedience to the principles of religion. Sometimes He descends personally, and sometimes He sends His bona fide representative in the form of His son, or servant, or Himself in some disguised form.
(Bhagavad-Gita-----4:7-----purport).

In two hands Lord Viṣṇu always carries a club and a cakra to kill demons, and in His other two hands He holds a conchshell and a lotus to give protection to His devotees. When His incarnation is present on this planet or in this universe, the Lord kills the demons and protects His devotees simultaneously. Sometimes Lord Viṣṇu appears in His person as Lord Kṛṣṇa or Lord Rāma. All of these appearances are mentioned in the śāstras. Sometimes He appears as a śaktyāveśa-avatāra like Lord Buddha. As explained before, these śaktyāveśa-avatāras are incarnations of Viṣṇu's power invested in a living entity. Living entities are also part and parcel of Lord Viṣṇu, but they are not as powerful; therefore when a living entity descends as an incarnation of Viṣṇu, he is especially empowered by the Lord.
(Srimad Bhagavatam-----4:19:37-----purport).

In his book Laghu-bhāgavatāmṛta, Śrīla Rūpa Gosvāmī has enumerated the following twenty-five līlā-avatāras: Catuḥ-sana, Nārada, Varāha, Matsya, Yajña, Nara-Nārāyaṇa, Kapila, Dattātreya, Hayaśīrṣa (Hayagrīva), Haṁsa, Pṛśnigarbha, Ṛṣabha, Pṛthu, Nṛsiṁha, Kūrma, Dhanvantari, Mohinī, Vāmana, Paraśurāma, Rāghavendra, Vyāsa, Balarāma, Kṛṣṇa, Buddha and Kalki.
(Sri Caitanya Caritamrta-----2:6:99-----purport).

There is a list of different incarnations of God. In that list Lord Buddha's name is also there. Lord Buddha's name is described: kīkaṭeṣu bhaviṣyati. Buddho nāmnāñjana-sutaḥ kīkaṭeṣu bhaviṣyati. In Śrīmad-Bhāgavatam the Lord Buddha's name is mentioned as future incarnation. Bhaviṣyati, "will appear." Kīkaṭeṣu, "in the province of Gayā."So this is called śāstra. Because Śrīmad-Bhāgavatam was composed five thousand years ago and Lord Buddha appeared 2,600 years ago... Therefore five thousand years ago Lord Buddha's case was in the future. Therefore it is said bhaviṣyati, "He will appear." This is called śāstra. Trikāla-jña. Śāstra writers, they are not ordinary men. Just like Kṛṣṇa is speaking. He is not ordinary man. Nobody will be interested so much if Bhagavad-gītā was written by ordinary man. It was spoken by the Supreme Personality of Godhead, and it was recorded by His incarnation, Vyāsadeva. So it is transcendental literature. Ordinary literatures, they cannot be perfect because there are four defect: bhrama-pramāda-karaṇāpāṭava-vipralipsā. Bhrama means "to commit mistake." Pramāda means "illusion," and vipralipsā means "cheating," and karaṇāpāṭava, "inefficiency of the senses." So śāstra means above these defects. Where there is no such defect, that is śāstra. And you can understand how five thousand years ago Lord Buddha's appearance was predicted. Similarly, still there is prediction about kalki-avatāra, which will take place about four lakhs and 27,000 years hereafter. Kalki-avatāra's name, his father's name and where he will appear, everything is there. This is called śāstra.
(Lecture on Bhagavad-Gita-----7:7).

Here, bhuvo bhāra-jihīrṣayā. An avatāra comes to do some particular work. Just like Lord Rāmacandra came to punish Rāvaṇa. That is avatāra. And there are many avatāras, they are mentioned in the Śrīmad-Bhāgavatam: Matsya-avatāra, Vāmana-avatāra, Kūrma-avatāra, and Varāha-avatāra, Vāmana-avatāra, Nṛsiṁha-avatāra, and Paraśurāma-avatāra, Balarāma-avatāra, Buddha-avatāra. Buddha is also avatāra. Keśava dhṛta-buddha-śarīra jaya jagadīśa hare. We do not agree with Buddha philosophy. Caitanya Mahāprabhu said, veda nā māniyā bauddha haya ta' nāstika. Buddha philosophy is atheism: "There is no God." He says "There is no God." So that is atheism. And especially... That there is no God, there is God—that is not the point. His point was to stop the animal-killing. Sadaya-hṛdaya darśita-paśu-ghātam. Kṛṣṇa became very, very, sympathetic with the poor animals. "Oh, in the name of yajña, these rascals, rogues, are killing so many animals." Therefore He came as Buddha-avatāra. Sadaya-hṛdaya darśita-paśu-ghātam. Keśava dhṛta-buddha-śarīra jaya jagadīśa hare. So Vaiṣṇava knows that "Here is Lord Buddha. He's Kṛṣṇa's avatāra," although we don't take his philosophy, because Buddharefused to accept Vedic authority. Nindasi yajña-vidher ahaha śruti-jātam. Śruti means Veda. Śruti-jātam. In the śruti, in the Vedas, there is mention of paśu sacrifice, animal sacrifice, there is. So they began to argue with Lord Buddha that "You are stopping yajña-vidhi, stopping animal-killing in the yajña. This is mentioned in the Vedas." But Lord Buddha knew, "These rascals simply talking of Vedas, he does not know what is Vedas. But what is the use of arguing?" Therefore he has to say, "I don't care for your Vedas. Stop animal killing." So nindasi yajña-vidheḥ. Yajña-vidhi, he wanted to stop. No more animal sacrifice in the yajña. Therefore he is denying the authority of Vedas. Nindasi yajña-vidher ahaha śruti-jātam. We cannot violate which is mentioned in the... Therefore he was taken... But he's avatāra. It was needed at that time.
(Lecture on Srimad Bhagavatam-----1:7:25).

So far Buddha is concerned, he's also considered śaktyāveśa avatāra. He preached this nirvāṇa philosophy. Although he did not speak about God, because it is considered that he was himself God, but the people amongst whom he preached, they were mostly atheistic people; therefore he did not preach about God. But he did not deny also. He simply wanted to make extinction of this present worldly activities. That was, yes... Nirvāṇa. And he represented the sacrifice of renouncement. He..., you may remember that the Supreme Personality of Godhead, out of His six opulences, one opulence is renouncement. So Lord Buddha's life is renouncement. He was prince. He, he was in a very young time. He renounced the world and underwent severe penances. These are the symptoms by which we can understand that he's also śaktyāveśa avatāra. And the Bhagavad-gītā you'll find, yad yad vibhūtimat sattvaṁ mama tejo-'ṁśa-sambhavam. Anyone, not only Lord Buddha or others, but anyone, Lord, in the Bhagavad-gītā it is stated, anyone who has got some extraordinary power, uncommon power, he's to be considered vibhu. Śaktyāveśa avatāra, there are two kinds, one directly empowered for particular mission, comes from the transcendental spiritual sky, and others, those who are in this material world, but they have got some specific power, uncommon power, not found in ordinary man. They are called vibhūti. This vibhūti (is) explained in the Bhagavad-gītā: yad yad vibhūtimat sattvaṁ mama tejo-'ṁśa sambhavam. That is out of the opulence of the Supreme Personality of Godhead.
(Lecture on Sri Caitanya Caritamrta-----2:20:367-84).