As evil in the world increases, Vishnu and other devas incarnate in full or in part to rid the world of them. The full avatars are Purnaavatar and the part avatars are Amshaavatar. The Mahabharata Adiparva Chapters 54-64 has a complete list. Below is a list from the Shrimahadevibhagavata 4th Skanda ()Deva – Avatar)
Kashyapa – Vasudeva
Adisesha – Balabhadra
Narayanarshi – SriKrishna
Yamadharma – Yudhishtira
Ashwinis – Nakula, Sahadeva
Dharma – Vidura
Shiva – Ashwathama
Gandharvaraja – Devaka
Ashtavasu (one of the ashtavasus) – Bhishma
Marudganam – Kripa, Kritavarma
Aditi – Devaki
Nararshi – Arjun
Vayu – Bhima
Surya – Karna
Varuna – Shantanu
Marut – Virat
Hansa – Dhritarashtra
Brihaspati – Drona
Dwapara – Shakuni
Pavaka – Dhrishtadyumna
Kali – Duryodhana
Rakshasa – Shikhandi
Varuna – Drupada
Dhriti – Madri
Viprachitti – Jarasandha
Hayagriva – Keshi
Bashkala – Bhagadatta
Lamba – Pralamba
Sanalkumara – Pradyumna
Lakshmi – Panchali
Sidhi – Kunti
Mati – Gandhari
Jaya – Hiranyaksha
Vijaya – Hiranyakashipu
Hiranyaksha – Ravan
Hiranyakashipu – Kumbhakarna
Ravan – Sishupala
Kumbhakarna – Dantavaktra
Prahlad – Salya
Kalanemi – Kamsa
Anuhlada – Dhrishtaketu
Khara – Dhenuka
See Avatar for more.