The term eulogy is actually from the Greek, and translates to something close enough to true words. By reason of pure definition, a eulogy can be given in honor or praise of someone who is living and still be called a eulogy. But generally, the term refers to a speech given at a funeral in memory of someone who has died.
Callimachus, the classical Greek poet, may have been the first to implement the eulogy (or elegy, as he called it) in honor of his late friend Heraclitus. A translation of this piece may be found here: https://www.bartleby.com/101/759.html
First transcribed in the 1400s as the Latin eulogium, today’s eulogies are the offspring of the antiquated elegy, and place a spotlight on the life and legacy of a person who has recently passed away. (Eulogies differ from obituaries, which are more like notifications that a death has occurred, like those found in a newspaper.)
The Greeks and Romans made great use of elegies in their writings, and this romantic, emotional style of expression continued to be popular even into the 1800s. Writers such as Alfred Lord Tennyson, Thomas Gray, Katherine Philips and W.H. Auden published elegies in honor of someone they had loved and lost.
Nowadays, elegies have evolved into eulogies, which are no longer just poems and which are spoken primarily at funerals or wakes. Eulogies can be still be customized, however, as poetry or as any style of writing befitting of the person who is being memorialized.
Click below to read some of the more well-known elegies of old.
~In Memoriam A.H.H. by Alfred, Lord Tennyson, 1849
~Elegy Written in a Country Churchyard by Thomas Gray, 1750
~Stop All the Clocks (or Funeral Blues) by W.H. Auden, 1938
~Epitaph by Katherine Philips, 1654