epithetsFormulaeMeterTLG

Epithets

Epithets are those short phrases that accompany the name of a person or the mention of a thing.

when Achilles is sitting in his tent, he is still "swift-footed" but in that context, the epithet is not particularly relevant. It identifies some trait of him.

oop, first, need to make sure everyone has a grasp of some basic facts about meter

the hexameter line has a series of "short" and "long" syllables. I'll use DUM for long syllables and da for short ones. So each syllable is the smallest unit of meter.
Next, the 'feet', also known as "metrical feet," are units that have more than one syllable:

DUM-da-da is called a "dactyl" (English "STRAWberry" is usually pronounced this way)

dactylos means "finger" in Greek, and the dactyl has one long and two shorts just as your fingers have one longer bone and two shorter bones.

DUM DUM is called a "spondee" (English "big dog" is usually pronounced this way)

To build a dactylic hexameter, you need six (hex means six) feet.
Mostly, you use dactyls, but spondees can substitute for most of those dactyls when that helps the poet.
So the line's shape is:

DUM¹-da-da-DUM²-da-da-DUM³-da-da-DUM⁴-da-da-DUM⁵-da-da-DUM⁶-DUM
But you can substitute a DUM-DUM spondee for most of those DUM-da-da dactyls, and the last foot is always DUM-DUM.

There are, of course, more details to all that, but that's what we need to look at how epithets work and why they are important in Homer.
But first, note that all songs in general have meter, and so does most poetry (can't speak for contemporary poetry, but I bet it does too). That's why Weird Al can parody "Lola" by the Kinks so effectively: he simply finds different words that match the same meter:

I met her in a club down in old Soho where you drink champagne and it tastes just like coca-cola, c-o-l-a cola, etc. (Lola by the Kinks, from the 70's, a bit after stonewall, about a cross-dressing (or maybe a trans-gender) person and a cis-het man
I met him in a swamp down in Dago-ba where it bubbles all the time like a giant carbonated soda, s-o-d-a soda, I saw the little runt sitting there on a log ... (Weird Al Yankovic's parody, about Yoda)

BACK TO EPITHETS: just like words that fit the rhythm of Lola, epithets have a certain shape that fit the rhythm of a Homeric line: some are DUM-DUM-DUM-da and so can fit the end of the line (or the beginning), some are da-DUM-da and so can fit in the middle, others have different shapes (da-da-DUM, da-DUM-da-da, etc.
A demonstration search the TLG for "Achilles" happens here (those who are not watching this live should go find the recording on the teams channel called " Homeric Epithets and the Thesaurus Linguae Graecae" and watch it, then come back and read from here on.
A man named Milman Parry, and his student Albert Lord, were the first to really realize that epithets worked as part of a fundamentally oral composition technique. Thus began "oral studies" of Homer.
It's not just epithets that work like this: for instance, when you kill someone in Homer, there are several verbs you can use (just as in English we have "kill," "slay," "slaughter," "cut down," etc.) and thus there is a variety of metrical shapes.

what's really cool about that is that the bard had a very economical system: even if there are several ways to say "he killed" that have the pattern DUM-da-da-DUM, to pick a random metrical shape, there is usually only one such form that is actually found in Homer per metrical shape. So if "he killed" is at the end of the line, it might always be "he slew," but if it's at another point in the line, it might always be "he slaughtered," and at another point in the line, "he brutally killed."
that makes these things predictable: the bard doesn't need to make a choice between "kill," "slay," "slaughter" etc.: the metrical slot comes up, and the one thing that fits it comes to mind, and it all snaps together at the moment.

All of these building blocks of the Homeric line are called formulae: there are formulaic epithets, but also verbal phrases, noun phrases, prepositional phrases, etc. It's one huge lego set with many shapes and sizes and varieties of blocks that work very efficiently together to build songs.
When an epithet has a fixed place in the line, one in which it consistently occurs, that is called a fixed epithet
So that's kind of cool, in it's own right, but what's the bigger payoff for this?
First, when we have a fixed epithet, one that doesn't seem particularly appropriate to the context, we don't need to worry about why it's there: Achilles can be "swift-footed" even lounging in his tent, and the poet wasn't trying to make some ironic comment or some other crafty comment in those cases. That was just the piece that fit.

But that does not make the piece meaningless or superfluous: The epithets invoke a whole world of Homeric verse each time they occur: rather than have a one-time context-appropriate meaning, whenever it occurs, "Grey-eyed Athena" conjures up every time Athena does anything, a huge network of what Athena is, as does "wise Penelope" for Penelope, as does any epithet for the person or thing it applies to.

Some parts of each line are creative opportunities, places where the bard can create unique phrases, that do not occur elsewhere.
But other parts are formulae.
So the bard blends together the creative with the automatic.
Likewise, the audience could easily predict how the poet would fill in the formulaic parts, but would find the creative parts more novel. It's a pleasing mix of the familiar, the repeated element with the more unique elements, perhaps even surprise elements.
If we combine all these small formulae with type-scenes, we can see that on the micro scale of words, there is creativity blended with formulaic tradition, and, ZOOMING OUT A BIT, on the middle scale of short type scenes, there is also creativity blended with more predictable tradition.

And, ZOOM OUT AGAIN on an even larger scale, the scale of larger scenes like the teichoskopia /view from the walls, which is made up of some unique things and some type scenes, we again have creativity blended with more predictable tradition.

SO each bard could be a unique creative artist, but also a craftsperson true to a massive and predictable tradition.

Some will have been more predictable, even entirely predictable, great crafters, perhaps not great innovative artists
Others might have been creative virtuosos: great crafters and great innovative artists
But most will have been pretty predictable, because a tradition is predictable, and that's what makes it a tradition.
It's a big mix of improv/inovative composition and performance and rehearsed, memorized performance: different bards surely had different mixes, and perhaps, just perhaps, there was one named Homer who was a virtuoso in some way (plot? voice? instrumental skill? phrasing? more than one of those?) and so his name got attached to this massive tradition and huge web of bards. Many people have a hard time imagining that even in an oral world, a poem like Iliad or Odyssey could form without an individual hand to shape it. I'm not at all sure of that.