There are three basic ways to tell a story, and none of them is really better than the others, so to speak. And this isn’t really supposed to tell you which one to choose, because I can’t decide for you, it’s strictly informative. :P
A story in the first person is told straight from a viewpoint character, as if the camera is sitting inside their brain and looking out their eye sockets. (Isn’t that a lovely thought?) First Person tends to be quite subjective; it’s a bit more objective if the viewpoint character is not the main character, such as in The Great Gatsby.
People speak in the first person when recounting what happened to them: “I bought a goldfish for my daughter the other day, but my cat is a little too enthralled with the thing.”
This is often seen as more immersing than a third person story, but this hangs heavily on the skill of the writer. First person tends to be spotted in young adult novels, so the reader can identify more with the viewpoint character, but this also depends on how skilled the writer is.
First person narrators by and large are a bit unreliable; events are skewed slightly depending on what the narrator did or did not see, and occasionally are tainted by the narrator’s opinions. Some are more reliable than others, however. Case in point: Ishmael of Moby Dick often tells things how they are, whereas Scout of To Kill a Mockingbird tells things how a young girl would see them. An honest character will be more reliable as a narrator than a character who bends the truth, omits some parts of the truth, or outright lies. A very young character such as Scout will probably miss some nuances in other people’s behavior, but an older character like Atticus would likely spot these and maybe internally make a comment about it. This can make for some interesting narrative.
Third Person Limited
This is when the “camera” is always hovering near the viewpoint character, and occasionally will enter his mind so we can read what he’s thinking. We don’t read the thoughts or emotions of the other characters unless the character happens to be psychic.
When you tell a story to your best friend or dog or something you’re referring to other people in it in the third person: “Mr. Van Damme bought a goldfish for his daughter the other day, but his cat is a little too enthralled with the thing.”
Third person limited, in the hands of a skillful writer, can be even more engaging than if the story were written in first person. In this editor’s humble opinion, Twilight should have been written in the third person, because Bella in the first person wins the award for Most Obnoxious Character In The Known Universe And Probably Many Bits Of The Universe We Haven’t Got Any Idea About. But I digress.
Third Person Omniscient
…is when the “camera” visits any of the characters at will. It freely pokes into their minds and draws back when it so desires, and can visit characters a million miles away from another character. It’s likely the most objective of the viewpoint types.
This would probably be how Zeus would tell his fellow gods a story: “Mr. Van Damme bought a goldfish for his daughter the other day, and she thinks it’s really cute but her dad is somewhat dreading what would happen to it if the cat got over its fear of water suddenly. His daughter isn’t too bothered though. Also the brother is secretly afraid of girls.” Zeus watches and laughs ‘cause he’s immortal.
But anyway, this one has been falling out of favor lately; you can see it more often in older works than nowadays. Supposedly it’s impersonal and whatnot. But it would make for a curious story if you combined first person and omniscience.
The moral of the story: use common sense.