Yes, Éire is the Irish-language name for the Republic of Ireland. The name derives from "Ériu," the goddess of Ireland in Celtic mythology. The English name "Ireland" comes from Old English "Īrland," likely from Proto-Celtic "Iverio" meaning "land of yew trees."
Constitutionally, the state is named "Ireland" (English) and "Éire" (Irish). The term "Republic of Ireland" is a descriptive designation often used internationally to distinguish it from Northern Ireland.