First of all, the term "point of view" is loosely used in two different ways when discussing writing craft.
One meaning is essentially equivalent to narrator, or protagonist. Note these are not necessarily the same thing. For example, in the Sherlock Holmes stories, Watson is the narrator but Holmes is the protagonist. In this sense, the question to be examined is at the intersection of whose story is being told, and who is doing the telling.
The other meaning of "point of view" addresses the topic of grammatical person. The overwhelming majority of fiction in the Western tradition is told in either first or third person, I believe with a tendency to favor third person narratives. For fiction purposes, we also talk about "tight", "close", "loose" and "omniscient" point of view. That is to say, story focus. So, for example, one might say that a text is in "loose third person". In this sense, the question to be examined is from what linguistic and stylistic perspective the story should be told.
With respect to choice of narrator or protagonist, a sensible default rubric is to determine whose story is being told by asking which character experiences change, transition, loss or personal growth. Absent other considerations, that's probably the character whose story you want to tell.
Note there are at least as many counterexamples to this as there are examples of it. How much does Holmes really change during any of Sir Arthur Conan Doyle's stories? Or even over the entire arc of the original canon? Likewise, in the movie version of P.L. Travers' Mary Poppins [ imdb ], it is Mr. Banks who is changed and experiences profound transition, even though the film is nominally concerned with the relationship between the title character and the Banks children.
There are plenty of very good reasons to tell stories with these equivalent of over-the-shoulder basketball shots, but the simplest and clearest way to approach the story is directly.
This decision intersects quite strongly with the concept of grammatical person and story focus. First person is of course an "I" story. Third person is of course a "he" or "she" or "it" or "they" story. The focus has to do with where the narrative equivalent of the camera sits. Tight stories are told from close inside the narrator's perspective. They literally can't see or know anything on the page that the story action doesn't show them. Most first person narratives are tight, for somewhat obvious reasons. A close narrative pulls back from the behind-the-eyes perspective and sits more behind the shoulder. Loose pulls back even further, following the character around from a distance. Omniscient virtually abandons the character for a wider-ranging view of the world.
Note that the looser the focus, the stronger the implied narrator. Stories written in loose or omniscient point of view can have a very strong voice as the implied narrator. Sometimes authors will explicitly acknowledge that implied narrator as a stylistic conceit, as in James Morrow's The Last Witchfinder, where the story is narrated by a copy of Newton's Principia Mathematica.
When you're working in tight or close point of view, the narrative will generally notice things the character can or would notice. It will use words and concepts the character would. What a trained assassin sees on entering a room is very different from what an interior designer sees. The way they would describe the space is very different. That in turn infuses the descriptive and expository prose being used in the scene.
On the other hand, in loose or omniscient point of view, the implied narrator can take over and make all manner of observations either explicitly or implicitly, use different speech registers, and take significantly different approaches to story telling than the character would.
To return to the concept of a default rubric, absent other considerations, in Western story telling traditions we tend to write in close third person in the simple past tense, sometimes referred to as the "narrative present".
I haven't really touched on choices of verb tense, application of tense shifts, different types of narrator, intrusions, and many other techniques that inflect point of view. This post barely scratches the surface. Like I said, it's a complex topic about which the more I learn, the less I understand. From my own perspective, my best work with point of view has been in my novellas "America, Such As She Is" and "The Baby Killers". I also did some work I'm pretty pleased with in the Green books. Frankly, I'd be at a loss to analyze any of those in credible, objective detail.
That being said, for a newer writer still exploring the fundamentals of point of view, I hope these touchpoints will be helpful.
I am quite curious what the writers and editors reading this blog think about point of view. Where did I get it wrong? How would you explain the concepts? Can you intelligibly go deeper than I am able to?