Most people think that Vishnu has only 10 avataras. However, the Srimad Bhagavatam says Vishnu incarnated 24 times. So who are the 24 avataras of Lord Vishnu?
It is explained in the Bhagavatam Canto 1 chapter 3. Even though there are innumerable incarnations of the Lord, some of the important ones are mentioned in this chapter.
The four unmarried sons of Brahmā [the Kumāras] also called as Sanat Kumāras.
Varaha avatar (Boar)
Devarṣi Nārada, who is a great sage among the demigods
Lord Kapila, foremost among perfected beings. He gave an exposition of the creative elements and metaphysics to Āsuri Brāhmaṇa.
Dattatreya, the son of the sage Atri. He was born from the womb of Anasūyā, who prayed for an incarnation.
Yajña, the son of Prajāpati Ruci and his wife Ākūti.
King Ṛṣabha, son of King Nābhi and his wife Merudevī.
King Pṛthu who cultivated the land to yield various produce, and for that reason the earth was beautiful and attractive.
Matsya, The form of a fish.
Kūrma, The form of a tortoise.
Nṛsiṁha - The man lion form.
Vāmana, the form of a dwarf brāhmaṇa.
Sri Rama, the son of Dasharatha.
Vyasa, the son of Parāshara.
Buddha, the son of Añjanā, in the province of Gayā.
Kalki incarnation and become the son of Viṣṇu Yaśā.
These are some of the important avataras mentioned in the Bhagavatam in the Canto 1 chapter 3. But his incarnations are innumerable and infinite.
अवतारा हि असङ्ख्येया हरेः सत्त्व-निधेर् द्विजाः
यथाविदासिनः कुल्याः सरसः स्युः सहस्रशः
avatārā hi asaṅkhyeyā hareḥ sattva-nidher dvijāḥ
yathāvidāsinaḥ kulyāḥ sarasaḥ syuḥ sahasraśaḥ
O brāhmaṇas, the incarnations of the Lord are innumerable, like rivulets flowing from inexhaustible sources of water.
Similar thing was said to Arjuna by Kṛṣṇa in the Bhagavad Gita.
बहूनि मे व्यतीतानि जन्मानि तव चार्जुन |
तान्यहं वेद सर्वाणि न त्वं वेत्थ परन्तप || ५ ||
bahūni me vyatītāni janmāni tava cārjuna |
tānyahaṃ veda sarvāṇi na tvaṃ vettha parantapa || 4.5 ||
Many births of Mine have passed, O Arjuna, and so is it with you also. I know them all, but you do not know them.
As per bhagavatam and as explained by Sriman Madhwacharya, who propagated Dwaitha matha, the 24 names of Sriman Naarayana, are , kesava, naaraayana, madhava, govinda, vishnu, madhusoodana, trivikrama, vamana, sridhara, hrushikesa, padmanabha, damodhara, sankrushna, vasudeva, pradyumna, aniruddha, purushottama, adokshaja, narasimha, achutha, janardana, upendra, haraye, srikrishna. These names of Sriman Naaraayana are chanted while having achamana during Sandhyavandana kala.