Damodar (also spelled "Damodara" and "Damodarah") is the 367th Name of Vishnu from the Vishnu sahasranama. The various meanings of the name are given as follows:
Celebration of Krishna as Damodar
A popular bhajan that celebrates Krishna as Damodara is the Damodarashtakam (found in the Padma Purana of Krishna Dwaipayana Vyasa, spoken by Satyavrata Muni in a conversation with Narada and Shaunaka.). It is often sung by devotees during the month of Kaartika, and is very popular amongst the Vaishnavas of ISKCON.
References
Damodar (name of Krishna) Wikipedia(Text) CC BY-SA