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.