The Hero's Journey is a framework that scholar Joseph Campbell came up with that many myths and stories follow. Many storytellers and story-readers find it a useful way to look at tale. (That's actually putting it lightly. Some people are straight-up obsessed.) Chris Vogler adapted Campbell's 17 stages of a hero's journey, which many screenwriters use while making movies. Vogler condensed Campbell's 17 stages down to 12, which is what we're using. Check out a general explanation of the 12 stages.
The story of Daedalus and Icarus doesn't fit perfectly into the Hero's Journey structure, but we're giving it a shot. As the gross old saying goes, there's more than one way to skin a cat. Here's how we've diced up the story:
Before their big adventure, Daedalus and Icarus are happily hanging out in King Minos's court. Sure, Daedalus is occasionally forced to do unsavory things (like build a maze so that King Minos could feed innocent people to the Minotaur), but overall, life is good.
Call To Adventure
Things change once King Minos becomes angry with Daedalus for helping Theseus slay the Minotaur. He imprisons the inventor and his son in the Labyrinth, which is gross and probably smells like a dead monster. Daedalus loses his freedom, and must use his inventing powers to escape! Go, go, gadget!
Refusal Of The Call
This stage doesn't really apply to the story. Right off the bat, Daedalus knows that he must escape the Labyrinth and the island of Crete. He never doubts the call of his own instincts to get the stink out of there.
Meeting The Mentor
Again, not applicable. Sorry, Shmoopers. Daedalus has no mentor, because he is Greece's greatest craftsman. In order to "meet the mentor," all he has to do is check-in with himself and brainstorm a great idea—which he does.
Crossing The Threshold
Strapping a pair of wings to his back, Daedalus is the first human to cross the threshold of the sky. He flies into the open air, and his wings successfully keep him aloft. Huzzah!
Tests, Allies, Enemies
Daedalus puts a pair of wings on Icarus, and the two take off. Daedalus keeps looking over his shoulder to make sure that his little guy is doing well: he's worried that Icarus will dampen his wings with seawater or melt them with the heat of the sun.
Approach To The Inmost Cave
Enjoying his newfound powers of flight, Icarus starts to push the limits of his wings. He ignores his father's advice to remain at a sensible height, and flies higher and higher, towards the sun. This isn't going to end well.
Icarus's wings melt. He plummets to the sea, calling out to his dad as he falls. Daedalus can't make it in time, and Icarus drowns.
Reward (Seizing The Sword)
If this were a myth with a happy ending, Daedalus would have caught Icarus, or at least rescued him from the sea. Unfortunately, this ain't no happy ending. The only "rewards" Daedalus gets are (a) a few of Icarus' feathers floating in the water and (b) the realization that great inventions can have terrible consequences, and that trying to overcome nature is sometimes a rotten idea.
The Road Back
Daedalus flies on, eventually landing in Sicily (an Italian island). When he gets there, he befriends King Cocalus, the ruler of the island. He rejoins court life, and begins to lead the kind of life he did as part of King Minos' court in Crete.
This is usually when the hero has to face one final struggle before settling into his new life. Here, the final struggle takes the form of King Minos, who travels to Sicily to find Daedalus. Fortunately, King Cocalus' daughters have taken a liking to Daedalus, so they kill King Minos with boiling water. Natch.
Return With The Elixir
At the end of this myth, Daedalus is a much humbler man. He has learned that trying to achieve feats best reserved for the gods (like flying) can lead to disastrous results. Daedalus even builds a temple to Apollo while living in Sicily, in order to prove his new respect for the gods.
This new humility is Daedalus' "elixir." To find it, he had to make a great journey and pay a dreadful price—the loss of his son.