Primary characters are the big ones: the hero/protagonist, the villain/antagonist, the main viewpoint character. These are the characters the reader identifies as the ones the story is about. They’re the ones with the biggest stake in the outcome of the story, and usually they’re the ones who have the largest impact on whether or not that outcome succeeds or fails. They’re the ones the readers mean when they say, “There’s this guy who wants to…tries to…needs to…has to…” even if the reader can’t remember their names.
Most stories have at least one of the three – hero, villain, main viewpoint. (There are a few stories, like Joanna Russ’s “Useful Phrases for the Tourist,” that don’t have any characters at all by normal standards, but they are rare and always a tour-de-force). If there’s only one character, that character is usually a combination protagonist and POV. If there are two characters, you usually have a protagonist/POV and an antagonist.
It’s possible to have a Man-vs.-Nature survival story in which two people survive the shipwreck or crash or whatever, and one is the protagonist while the other is the POV. In that case, though, the two survivors generally start off hating each other, making it a protagonist/POV and antagonist situation again. This is because it makes a much more interesting story to have two people who hate each other, but who have to cooperate for mutual survival, than it does to have two friends who are cooperating. Alternatively, you have the two friends, but one of them is gravely injured, which adds time constraints to the survival problem, but frequently reduces the number of primary characters back to a protagonist/POV plus secondary injured/unconscious character.
The arguments over this particular way of classifying characters usually arise when the viewpoint character isn’t the protagonist. Dr. Watson is the narrator and viewpoint in most of Sir Arthur Conan Doyle’s Sherlock Holmes stories. Holmes is clearly the protagonist, but is Watson also a primary character, or is he just a Sidekick (a major character) who happens to be the POV?
My take is that being the viewpoint makes it much more likely that a character is a primary character, even if one would normally call them a major or even a secondary character based on their role in the story. The reason is that the reader gets to see into the head of the viewpoint character, and views the story events through the POV’s eyes and reactions. Every character is the protagonist of his/her own story; the minute they are given the viewpoint position, whatever story the writer is telling starts to wrap itself around the POV’s concerns, to become the story in which the POV is the hero. Sherlock Holmes is a strong enough and obvious enough protagonist that Watson never manages to promote himself to hero (and neither Watson nor Conan Doyle try), but Watson is still more than just an important sidekick.
The other argument comes when a story has an ensemble cast and the writer is using multiple viewpoints. This doesn’t guarantee that there will be more than three primary characters (I talked a few posts ago about the case of the secondary character who gets one viewpoint scene and immediately dies), but it makes it more likely. It also makes it hard to tell the difference between major ensemble characters and the primary characters.
This can create problems for the writer. Even in an ensemble structure, there’s usually one character the writer intends to be the hero/protagonist, but the more primary characters the writer has included in the rest of the cast, and the better the job the writer has done with making them primary characters, the more difficult it can be to tell which of the three or four central characters and/or plotlines the writer intended as the “main” story. Primary characters have individual stories and personal complications that weave in and out of the main plotline, and if one of them catches the reader’s sympathy and interest, the reader can mis-identify that character as the protagonist, and then get frustrated when the main plotline veers away from the story the reader was really interested in.
It’s also really difficult to juggle too many characters and subplots and still do them all justice. Using a multiple-viewpoint structure is one thing; allowing every POV character to grow into a primary character with his/her own story to tell can dilute the impact of the whole, unless the author is tightly in control and can keep all the various sub-stories coming back around to focus on the main plotline.
The two-primary or three-primary character story tend to have the opposite problem. When writers have only one POV character, they limit themselves to showing/seeing/telling only through that character’s eyes, reactions, and knowledge. Since the POV character is nearly always one of the Good Guys (if not the protagonist), this makes it really difficult to make the villain/antagonist a well-rounded, complex character, especially if, like Moriarty, the villain remains off-stage until the very end of the story.