Coward comes from the Old French word coart, from the Latin coda or cauda, meaning “tail (of an animal).” As a result, the word likely came to imply fear in a metaphorical sense—an animal’s tail tucked between its legs. Those familiar with musical notation will also recognize coda as the word for the concluding—or tail-end—passage or verse from a musical composition.
Coart, also spelled Kyward or Cuwaert, was the name of a cowardly hare that appears in some Old French versions of the folk tale “Reynard the Fox,” one version of which you can read in English here.
The suffix -ard (or -art) also comes from Old French and is typically pejorative or derogatory, as seen in the words braggart, drunkard, buzzard, and bastard†.
The word coward can also be used to refer to a bully or someone who uses their superior strength or power as an advantage over someone weaker or less powerful, regardless of whether they show tail-tucked fear when doing so. The entry on this word from A Dictionary of Modern English Usage (1926) by Henry Watson Fowler expounds upon this:
The identification of coward & bully has gone so far in the popular consciousness that persons & acts in which no trace of fear is to be found are often called coward(ly) merely because advantage has been taken of superior strength or position; such action may be unchivalrous, unsportsmanlike, mean, tyrannical and many other bad things, but not cowardly.
† Interesting side note: One theory about the origin of the word “bastard” is that it came from the phrase *fils de bast*, meaning “packsaddle son,” with the pejorative ending *-ard*. Saddles often doubled as beds while traveling, so the phrase implies a child conceived on an improvised bed away from home. Another suggested origin is that it comes from the Proto-Germanic banstiz, or “barn,” suggesting low origin.