Third Person (Limited Omniscient)
The narrator in this story is outside of the action, narrating from a distance. He can get inside the heads of some of the characters when he wants to, but he doesn't spend much time there. Take the very last sentence, for example:
She kept watching him even when she was through cutting the onions and she kept on watching until it was no longer possible for her to see him, because then he was no longer an annoyance in her life but an imaginary dot on the horizon of the sea. (13)
Here, we get just a little taste of Elisenda's motivations and thoughts. But mostly our narrator makes us get by with outward descriptions of what people say and do. This point of view makes it that much harder to figure out whether or not the title character is an angel or just an old man with wings—and what he's doing here in the first place.
Because of this outside view, we're put in the position of the other characters in the story. What you see is what you get, and, since the old man doesn't speak the same language as them, there's no communication for us to overhear.
Not So Nice
But, just because the narrator is outside the action doesn't mean he's nice. Describing the different reactions the townspeople have to the angel, the narrator says:
The simplest among them thought that he should be named mayor of the world. Others of sterner mind felt that he should be promoted to the rank of five-star general in order to win all wars. Some visionaries hoped that he could be put to stud in order to implant on earth a race of winged wise men who could take charge of the universe. (5)
Something tells us the narrator doesn't really consider any of these people "visionaries." In fact, they all seem kind of silly. This irony tells us that we should maybe be a little critical about the treatment of the angel—and maybe we should be careful about doing the same thing.