To eulogize someone is to praise them very highly, in speech or in writing. The strongest, most typical use is at a funeral: you eulogize the person who has died. People also eulogize a book, a leader, or a whole time in the past, meaning they talk about it with almost ceremonial praise. It is a formal verb. In everyday talk, “praise” or “speak highly of” is more common. American spelling is eulogize; British spelling is often eulogise. Though actually, you can eulogize the living too — a glowing tribute is still a kind of eulogizing — but the funeral picture sits behind the word.