Grammatically, first person just means "I" instead of he or she. But in fiction, if all you're doing is changing the he's to I's, then you're doing it wrong.
First person is a chance to get inside a character's head. Done right, the reader will identify with that character strongly, feel what they feel. The reader will get to know them more personally than with other POVs; they will see the world through their eyes.
First person also has the advantage of feeling more truthful. The narrator is involved in the story -- they were there when it happened -- so it feels less like fiction and more like an eyewitness account.
What makes first person work can also be limiting. For example, the reader only knows what the narrator knows and only sees what they see. Depending on how you want the tension presented, this can take some planning. Also, first person is inherently a flashback. This isn't so much a limitation as something to be aware of. If the main tension is that the narrator might die, well, that tension is gone every time the reader remembers that the narrator is telling the story.
Some tips on writing in first person:
- Find the narrator's voice. The biggest thing first person has going for it is that you get to speak in the voice of a character all the time. If the narrator's voice is just a third person narrator who says "I" instead of "he," it's almost a waste of the POV.
- Know why the narrator is telling the story. Are they trying to vindicate themselves? Keep others from making the same mistake? Tell their side of what happened? It doesn't have to be a unique reason, nor does the narrator ever need to say it explicitly, but as the writer you should know what it is.
- Know who the narrator's audience is. Like above, it doesn't have to be stated explicitly, but you should know who the narrator is telling the story to.
- Don't slip into other character's heads. WRONG: "I watched Nora from across the room. She was upset -- worried about the upcoming deal." BETTER: "I watched Nora from across the room. She looked upset -- probably worried about the upcoming deal."
- Don't show what the narrator is doing without ever getting into their heads about why. This is just as bad as not giving the narrator a voice.
- The narrator should be present at all major events. Otherwise the narrator might only hear about the climax from a friend, which is lame. A corollary to this is that the narrator should be active at the major events, not just a bystander.
DONE WELL: Fight Club, Chuck Palahniuk.
ARGUABLE: Beach Road, James Patterson (I liked it, but others who saw the twist coming, or who just got mad, didn't).
DONE POORLY: "The Stillborn Dead", a short story by me, which you will never read (failed because the narrator's secret was lame and/or didn't make sense).