First Person (Peripheral Narrator)—The Journalist; Third Person (Omniscient)
The Short Version
The frame story is told by a first person narrator (the journalist), and he's on the edge of Calvin's story; so it's first person, peripheral narrator.
The stories themselves are told by a third person narrator (no one in the story) that floats around pretty freely most of the time, able to peek into anyone's head; so it's third person, omniscient. Except it doesn't always seem omniscient in the usual sense. Which leads us to…
The Longer Version (Including "What Effect Does That Have on the Book?")
When Asimov wrote the frame story (the journalist interviewing Susan Calvin), he added the journalist as a first person narrator. So the book starts and ends with pretty clear "I" statements: "I looked at my notes" (Introduction.1); "I never saw Susan Calvin again" (Evitable Conflict.230). In between those two statements, the journalist gets very few lines. So why add the journalist to this book at all?
Well, maybe it's because the journalist character is in the same position as the reader: we're trying to figure out the truth about Calvin and the robots. Or maybe it's because the journalist helps to stitch together these stories into a cohesive narrative by responding to them. For instance, he reacts with "a sort of horror" to the idea that Byerley was a robot (Evidence.300). So, when we say that the journalist is only the peripheral narrator, we mean that he doesn't take part in any of the stories that Calvin tells him; but he may be an important part of how we read this book.
But the journalist is only in the frame, and Susan Calvin tells him most of the stories. However, you probably have noticed that that makes very little sense: Calvin tells stories that don't involve her (she's not on Mercury with Powell and Donovan for "Runaround"); and she tells about things that she couldn't possibly know. For instance, in "Robbie," Mr. Weston mutters something to himself (Robbie.138). How could Calvin know that? She couldn't know all the things that the stories tell us.
No, what we have here in the stories is straight-up third person style: the stories are told by a narrator who isn't a person in the story (or not even a person at all). But what kind of a third person narrator is it? Is it omniscient—knowing all, including what all the people are thinking? Is it limited omniscient—knowing a bunch, including what one or two people are thinking? Or is it objective—only telling us what we could see and hear if we were there and never telling us what people are thinking?
We've actually had a bit of an argument here at Shmoop over this (no fatalities, but several injuries—we take our literature seriously). And here's why: the narrator basically floats wherever it wants to; for instance, "Escape!" pops back and forth between Powell and Donovan on the spaceship and Calvin and Brain on Earth. The narrator also occasionally goes into multiple people's heads, even just for a peek. For instance: "'I see,' said Robertson, who didn't" (Escape.27). That's pretty omniscient, telling us that Robertson doesn't understand, even when he says he does.
However, although the narrator has moments of being omniscient, we think that most of the stories are told in a limited omniscient POV, focusing on Calvin. For instance, "Liar!" gets into Calvin's head more than any other character, like when Milton Ashe tells her he's marrying someone else.
And there's more. Even though the narrator occasionally tells us what a character is thinking or feeling, the narrator often just tells us the objective facts. For example, in "Reason," Donovan is shocked and angry that Cutie is spreading his religion. But the narrator doesn't tell us that Donovan is shocked and angry; the narrator tells us this: "He came charging down upon them, complexion matching his hair and clenched fists beating the air furiously" (Reason.80). That's just the objective facts of the situation, without looking into anyone's head.
So, we're calling this an omniscient third person narrator, but notice how the narrator sometimes leaves things out and just tells us what we could see, like an objective narrator. In other words, we could say that the narrator treats us like a scientist: we see the external facts of the world and we have to figure out what it all means.
But there's more—and don't worry, this will be the last thing: the narrator may tell us what the humans are thinking or feeling, but never tells us what a robot is thinking or feeling. Because the robots are really the main mystery of these stories—they're what we have to figure out.