The term "encomium" has several different definitions. It is Latin by way of the Ancient Greek word encomion, and in broad stroke definitions, it is the formal practice of writing or speaking words in praise of someone or something. Such praise has been written about fictional people too, and about animals, so the definition requires some fleshing out.
In Ancient Greece, there were essentially two types of encomium. One was songs or poems composed by poets in praise of a specific thing. The other was a specific literary device used by rhetoricians and teachers of rhetoric. Students who studied rhetoric in schools led by the Sophists, or by the competing schools of Isocrates and Aristotle, learned how to both create and deliver praise in formal form, just as the modern student of writing today learns how to construct essays and speeches. This tradition continued with Roman rhetoricians and poets, and people will still find them today, some delivered and some written on a variety of subjects. Often, they are considered distinct from eulogies, because they praise living people.
One of the earliest and most striking examples from Classical Greece is the Encomium of Helen attributed to the Sophist Gorgias. Actually, as most Sophistic writing goes, the praise of Helen serves an entirely different purpose than really praising the woman (of Trojan War fame). Instead, it serves as a defense of Helen, not really the intent of encomiums in general, and becomes praise of the ability of language to persuade. Helen cannot be blamed for running off with Paris because she was either persuaded by his wonderful ability to speak or she was under the influence of powerful love.
In fact, Gorgias’ methods of praise for Helen are roundly and perhaps justly criticized by the great writing teacher Isocrates, in his work called Helen. His work again fails the mark on praising Helen and attacks Gorgias for advocating using speech as a powerful drug that can persuade people to act against their own interest. Neither work sufficiently serves as true praise of Helen, but both works serve as praise of language and thoughts on how rhetoric should be used. Many encomiums are more straightforward, such as St. Paul’s writings on the topic of love.
A speech delivered about an actor receiving a lifetime achievement award usually doesn’t diverge into a speech on something completely different. Poems celebrating the beauty of a person, place, or thing can be open to multiple interpretations but often keep strict focus on the praise of something. Articles that are a send up of someone’s life and work, mostly someone still living, present the modern day rhetorical encomium. People need only look as far as superdelegate endorsements of presidential candidates to see a living and breathing example today.