At its simplest, POV is about whether the main character of a story is referred to as I, you or he/she (or conceivably it), using the grammatical concept of person. Referring to the speaker is known as first person, referring to the person or thing addressed is second person, and referring to anyone or anything else is third person. In some languages, like Latin, this is built into the grammar, and the persons have to be used in the correct order in the sentence.1
Most stories are written in first or third, with second being kept mainly for experimental work or choose-your-own-adventure books. It isn't easy to do well, though it can be highly effective in the hands of a master, such as in Italo Calvino's wonderful novel If on a Winter's Night a Traveller.
There's more to POV, though, than which person it's in. It can, quite separately from this, be limited or omniscient, deep or shallow, distant or immersive. And it's important to remember that these aren't absolutes, but sliding scales where specific stories (or even specific scenes in a story) are defined by positions on numerous axes.
It's also important to bear in mind that none of these POVs is wrong. Some are unfashionable and will be much harder sells to editors or agents (second person is certainly one of these) and some are easier than others to get wrong. All POVs, though, suit specific stories can be written well, and if so are perfectly valid.
First person is increasingly popular, especially in YA fiction, although it's never been uncommon. There's a popular idea that a first-person story has to represent a supposed memoir by the character, or that they're telling the tale to someone. This is certainly an option, but by no means the only one.
First person narratives can range from immersive to distant. In the former case, the character is essentially passing on the experiences as they're happening, and will only write about what they're experiencing or thinking at the time.2 It's becoming quite common now to reinforce that with present tense, though past tense narrative is familiar enough not to feel wrong in such a context — I think of past tense in this situation as events being processed a second or two after they've happened.
At the other extreme, the character can be looking back on past events, whether writing, telling or just remembering it. This sacrifices the immediacy of the immersive approach, but it allows the narrator to reflect on their actions and introduce elements of the story or setting they couldn't have known at the time.
There are positions in between these extremes, and it's also not essential to write an entire story from the same position. In At An Uncertain Hour, for instance, I used the technique of switching between action the main character is immersed in as it happens and memories of his past life, written in a more distant style. I'm by no means the only author who's written like this — check out Iain Banks for a master of the technique.
Third person, where every character is referred to as he, she or it, is perhaps the most natural style of storytelling, but it comes in three broad forms: objective, omniscient and limited.
Limited third is perhaps the most popular POV today, although its popularity is maybe exaggerated somewhat by authors and editors. This is where, in any given scene, everything is filtered through one specific person's perceptions and inner thoughts. The greatest sin, if (but only if) you're writing in limited third, is head-hopping, where the POV changes from one paragraph to the next, or even within the same paragraph. Even this can be done effectively, but only if it's deliberately calculated for a specific effect, rather than to make life easier for the author.
The POV may be limited, but that can be handled in a number of ways. It can be deep and immersive, to a point where it's almost indistinguishable from immersive first person, or the "camera" can move out to show us details that aren't being directly noticed by the character. At its shallowest, for instance, you might describe the character's appearance as if from outside — without resorting to the dreaded mirror scene.
That level more or less merges into omniscient, which is where the POV is the author (or at least an authorial presence) who can show us whatever is necessary to tell the story, including the thoughts of multiple characters, general facts about the setting, and pithy comments about life, the universe and everything. It's distinct from head-hopping, though, because the revelations made by an omniscient narrator are made from outside rather than inside, reminding us that this isn't the character telling us what they're thinking, but the narrator knowing it.
The most extreme omniscient approach is the storyteller style, where the author is a direct presence addressing the reader. This is particularly common in a traditional style of children's story, where the author might interrupt the narrative to say something like "I expect you're wondering how he's going to escape from this. Well…"
Occasionally, the omniscient POV can also be a character within the story, usually playing a minor role, who either for specific reasons or just as a device has access to all the necessary information about the story and its setting. This character can be presented as either first or third person.
Objective is superficially a little like omniscient, since the POV is the author, but where omniscient is an active POV, objective is passive. Here, we're only told what can be seen and heard, not what any character makes of it or what they're thinking. Not much fiction is written this way nowadays, partly because it's incredibly difficult to write it effectively (I know, I've tried) but the mediaeval Icelandic sagas3 handled it brilliantly, using it to give a kind of dead-pan insight into the behaviour of the characters by inference, not by revelation.
This doesn't mean that a writer has to choose one of these precise positions and stick with it fanatically. Many writers who use limited third, for instance, will vary the depth from scene to scene, depending on what's needed at the time. Others operate right on the edge between shallow limited third and omniscient, sometime straying to one side of the border, sometimes to the other — the Harry Potter books are an example of this. The key, as with so much in writing, is always to know precisely what you're doing and why. It's possible to make your variations seem natural, rather than careless.
Who Should Be Your POV?
Some stories can be told exclusively through one pair of eyes. This is the default for first person, and many limited third stories also don't need more than a single POV, if the story is about that person. Other stories aren't about any specific individual and need a wide range of viewpoints to show the reader everything that feeds into the edifice the author's constructing.
George R.R. Martin is one of the best-known authors using a POV cast of thousands, but many stories require three or four POVs to cover everything. These POVs (usually in limited third person, though multiple first person novels are found4, and even switching between first and third for different POVs) will change either at the beginning of a new chapter, or else at the beginning of a discrete scene within a chapter. The technique is completely different from head-hopping.
A common piece of advice in choosing which character stands as your POV for a scene, a chapter, an entire novel, is that it should be the person most involved in what happens. That's often good advice, but not always. Sometimes, the person most genuinely involved in a scene, in the true sense, may be standing back and watching events unfold. Someone else may take centre-stage, but this is the person whose motives and assumptions are being challenged by what happens, which will affect their role in the rest of the story.
An extreme case is when the story is most effectively told by an observer, giving us a separate mind to filter the action through. The best-known examples of this are Dr Watson in the Sherlock Holmes stories and Nick Carraway in The Great Gatsby. Though not at all uninvolved, Watson stands a little back and provides us with the eyes to watch the enigma that is Holmes, while Carraway allows Fitzgerald to avoid having to reveal too much about Gatsby.
In the end, the choice of POV — the character, the pronoun used, the limit or omniscience, the immersion or the distance — is all about how to tell the story best, and that will be different every time. As it should be. The point of being a writer isn't to endlessly tell the same story.
1 This caused some problem for Henry VIII's minister Cardinal Wolsey, who grammatically but undiplomatically referred in his Latin letters to "ego et rex" (I and the king). This gave ammunition to those who accused him of arrogance.
2 In fact, there's a little wiggle-room here, since applying that too literally results in the extreme immersive first person style known as stream-of-consciousness — a valid approach, but one that isn't suitable for the majority of first person stories. As long as what you write is more or less in the moment, readers will normally ignore any slight discrepancy.
3 No, the sagas weren't oral tales told round the fire of a mead-hall. They were actually sophisticated literary works, produced by and for a surprisingly literate society.
4 Disclosure: I'm writing one myself at the moment, and finding the process fascinating.