Here's the incredibly brief version: A Wizard of Earthsea is a coming-of-age story. We meet a young boy who is wild and proud, then we see him make a terrible mistake and face something that he's not powerful enough to deal with. Once he's grown up a bit, we see him try to fix the terrible mistake he made.
And that's that! Wait, you're intrigued? You want more? OK, then, here goes…
The slightly less brief version: a wild boy is born, doesn't like his dad's work, and learns some magic from his witchy aunt. The boy uses that magic to defend his town when raiders attack.
Because of his success with the raiders, the boy is taken in as an apprentice by the mage Ogion, who gives the boy his true name, Ged. Ogion tries to teach Ged about magic and the balance of powers. But Ged wants more power (partly to impress a girl…you know how it is), so he almost summons a shadow monster. Luckily, Ogion stops Ged and offers to send him to school.
Ged goes to the magic school on Roke where he meets a friend, Vetch, and a rival, Jasper. Ged is very proud and, to shut Jasper up, he tries to summon a dead spirit. Instead, he succeeds in summoning a shadow monster. The Archmage of Roke saves Ged, but dies after banishing the shadow from Roke. Clearly this didn't go as planned.
Ged feels terrible, but Vetch is still his friend, so that makes him feel a little better. Ged continues to study magic, and after he graduates, he gets a little job watching out for dragons in a small town. See, after being traumatized by the shadow monster, Ged is way less fatheaded and less annoying.
But then, when he tries to save a friend's dying child, Ged runs into the shadow and he realizes he's in danger. He can't stay where he is because then he puts everyone in danger. He decides to fight the dragons first, and then go deal with the pesky shadow monster.
So he defeats the dragons (it helps that he knew the big bad dragon's true name). Then he tries to find a way to deal with the shadow. He goes to find a magic sword that someone told him about. But Ged was deceived and, instead of a good sword, Ged meets an evil magic stone and has to deal with that now too. It's not pretty, and Ged only escapes after turning himself into a falcon.
Ged flies back to Ogion, who helps him return to human form. Ogion also advises Ged to start being the hunter rather than the hunted. Where would Ged be without Ogion?
So Ged starts chasing the shadow and the shadow is scared. Ged is momentarily a castaway on an island where he gets part of a broken ring. (Which is important in the second book in the series, by the way.) Then Ged sails to an island where Vetch is, and the two of them go hunting for the shadow.
Eventually, Ged corners the shadow and absorbs it since it's really his shadow – it's part of him. Then they sail home now that Ged has: 1) fixed his mistake and 2) gained a deeper understanding of himself.