First person is when the narrator is a character in the story.
This is when the main character is the person telling the story. In other words, this is the "I" narrator. Examples: Holden from Catcher in the Rye or Katniss from The Hunger Games.
First Person Peripheral
This is when the narrator is a supporting character in the story, not the main character. This is still the "I" narrator, but now the narrator is not the protagonist. Example: Nick from The Great Gatsby (Gatsby is the protagonist).
Third person is when the narrator is NOT a character in the story.
Third Person Limited
Third person is the "he/she/it" narrator. Limited means that the POV is limited to just one character. This means that the narrator only knows what that character knows, only sees what that character sees. Examples: A Christmas Carol by Charles Dickens (where the story follows Scrooge at all times--even scenes that Scrooge would not be privy too we see through his eyes as he travels with the ghosts of Christmas past, present and future). The Book of Three (first book of the Prydain Chronicles) where the narrator follows the protagonist Taran.
Third Person Multiple
Again, we're in the "he/she/it" category, but now the narrator can follow multiple characters in the story. The challenge with this POV is making sure your reader knows when you're switching from one character to another. A good way to make the switch is to use chapter breaks or section breaks to signal a new POV. Example: The High King (which is the final book of the Prydain Chronicles) where the narrator follows several characters in the story, including Taran.
Third Person Omniscient
This one still uses a "he/she/it" narration but now the narrator knows EVERYTHING in the story. The narrator isn't limited by what the POV character knows. It's sort of like the narrator is god, hence the term "omniscient." This type of POV was very popular back in the day but has recently become less popular (some people feel like it's a little old-fashioned). Still, some excellent books use this narrator. Examples: Oliver Twist by Charles Dickens, and Pride and Prejudice by Jane Austen.
Third Person Objective
Just like the omniscient narrator can get into any character's head, the objective narrator gets into NO ONE's mind. This means the objective narrator can only relate information that is easily visible (character's words and actions). This narrator can't tell us about the character's thoughts or feelings because it doesn't know. It's kind of like watching a movie, where the only information you get is what you can see or hear. This POV is very tough to sustain for long pieces which is why the only example I can find is a short story: Raymond Carver's Little Things.
Other POV Choices
This is the "you" narrator. "You go to the store and realize you forgot your wallet... etc." Like objective POV, the second person is hard to sustain so there are very few novels written in second person. This POV is more popular for short stories. In fact, the first story I ever published is in the second person (which is weird because I think it's the only story I've ever done in second person). Anyway, if you're curious, it's here.
Unreliable First Person
This is when you have a first person narrator but you can't trust him/her for any number of reasons. Maybe the character is a very young child who doesn't really understand what's happening in the story. Or perhaps the character is insane. Or better yet, the character could be perfectly sane but also a pathological liar so you can't believe what she says. Example: The Tell-tale Heart by Edgar Allen Poe.
Epistolary (or other) Form
Epistolary is when the story is told in letters. There are many forms that work similarly to epistolary forms, like journal form or a story told through emails, etc. Mostly these forms work like the first person because the main character in the story is often the one writing the letters/journal/etc. The difference is that the story is limited even further because of the form. For example, people don't usually write dialogue in their letters, so if you want to use dialogue in epistolary form, you'll have to find a way around that.
In the end, POV is all about consistency. Whatever form you decide on, it's important to let the reader know what the "rules" are for your story and then stick to them.
Edit: Added and corrected a few examples.