A story told in first-person perspective is being told by one of the characters in the story, usually the central protagonist. Here's an example:
I closed my eyes and jumped off of the roof, silently praying that Dall was as good a pilot as he claimed. The duke shouted in alarm, screaming for his men to stop me, and then his voice was drowned out in an angry rush of wind.
I had been determined to keep my eyes closed as I fell, in hopes that I would be less afraid if I couldn't see the ground approaching. But it turns out that not knowing how far away the ground is was far more frightening than knowing. My eyes popped open after only a few seconds of falling.
The ground wasn't nearly as far away as I would have liked, and for a brief moment I panicked; then Dall's skimcar appeared from around the side of the building. Dall waved at me from the cockpit as he pulled up alongside me, matching my pace. I reached out and grabbed the top of the car, my fingers clenching so hard that I thought they might actually bend the di-steel.
Dall slowly maneuvered the skimcar beneath me and then slowed our descent. He slowed more abruptly than I would have liked, and I thudded into the cockpit with a grunt, spread-eagled. It wasn't bad enough that it hurt, but it did knock the wind from me.
"Sorry, Joan" Dall said with that irritating grin of his, his voice muffled by the glasstic between us.
"Just let me in!" I shouted.
He shook his head. "Not until we come to a complete stop. Too dangerous otherwise."
"Too dangerous?" I shrieked. "I just jumped off a three-hundred story building without knowing for sure that you were down here, and you think that would be too dangerous?"
I should take a moment to mention that I'd had a little problem with heights ever since I was six years old, and Dall knew that perfectly well. He was just being a jerk.
"You'll be fine out there for a minute or two," Dall replied, still grinning.
In the case of this example, one of the characters—Joan—is telling the story, as though she were sitting in front of us speaking. She uses "I" to refer to herself, marking this as first-person perspective.
So, what are some of the advantages and disadvantages of first-person perspective?
Intimate: Because the character is relating the story themselves, first-person perspective often creates a more close, intense, and intimate feel than other perspectives, and the action often feels more immediate.
Strong Personality: When a character is relating a story "in their own words," so to speak, it can often be much easier to get their personality across than it might have been in another perspective.
Unreliability: If you want to tell a story with an unreliable narrator, first-person perspective is often the way to go. Since the perspective character is telling the story, their opinions and prejudices can color the narrative, or they can even outright lie to the reader. This can make for some very interesting stories.
Info Dump: It's easier in first-person for your narrator to step away from the story for a moment to explain something that the reader needs to know; Joan did it near the end of our example.
Simplicity: First-person can often be one of the easiest perspectives to write without making perspective errors.
DisadvantagesMost limited perspective: Since one of the characters is telling the story, the author cannot relate anything that the main character cannot see. If you want your audience to know something, then your main character has to know. You can't just jump to someone else's head for a bit and then come back. (There are stories with multiple first-person perspectives, but these can be very tricky to write well.)
Difficult to explain character appearance: After all, if I were sitting and telling you a story, I wouldn't need to explain what I look like, would I? Look back over our example; there's not any information there on what Joan looks like, and it would be very awkward to try to fit in a description of her anywhere.
Difficult to make the audience worry for the character: Since first-person perspective requires the main character to be alive and well to tell the story, it is difficult to get your audience to worry about the safety of the narrator. How would the story be told, after all, if the main character weren't around to finish it? This is particularly true of first-person stories told in the past tense.