A "first person" narrator is a narrator who is also a person. You know the narrator is a person because he or she uses pronouns like "I," and "me." By contrast, a "third person narrator" is not a definite person, but usually a disembodied voice of unknown origin. So, the unnamed narrator of "The Black Cat" is obviously a "first person" narrator. He's a "central narrator" because he's talking about things that he did or things that happened to him, rather than things he watched, or heard about.
Like many Poe narrators this one is unreliable. This means he gives us reason to doubt one or more aspects of what he tells us. We are put on the defensive from the first lines of the story, when the narrator says he doesn't "expect" us to believe him, and that he won't even ask us to (1).
If we try to figure out if the narrator is telling the truth, we might fall into the story's dark and bottomless trap. If we think the narrator is lying about, say, the image of the gallows on the second cat's fur, then is he lying about all the abuse and murder as well? Without any outside perspective, it's all or nothing. If we try to sort truth from lies we dismantle the entire story.
As such, a helpful way of approaching the narrator's unreliability is by looking for what might be left out of the story, or what the narrator misses, but the reader sees. To that end, watch out for this guy's fancy language. The more carefully we read, the more the narrator actually reveals to us. If we go too fast, he loses us in his web of words.
For example, the man admits freely to horribly abusing his wife. During the worst of this he describes her (chillingly) as "uncomplaining" and "the most usual and the most patient of sufferers" (22).
Now, "uncomplaining" and "patient" are fairly obvious, but "usual" has a double meaning. It means that she's the narrator's "usual," or most frequent, victim. It also means that she suffers in the "usual" way, i.e., crying, screaming, etc. So, the man might be slyly admitting that he's being sarcastic when he describes his wife as silently and obediently submitting to his abuse.
But, maybe he's not being sarcastic. Maybe she was that traumatized, and found silence and submission the safest way to deal with him. The point is, the narrator dehumanizes her in his story, treating her like a thing. He leaves out her point of view, almost completely. You can take a look at her "Character Analysis" to see what we've done with the little information we have on her.
Poe wanted his stories to help readers exercise their analytical skills, and was fascinated by the idea of "secret writing" which you can read about here. An unreliable narrator helps keep us awake and on the lookout for errors, inconsistencies, and improbabilities, and invites us to read actively, and to openly challenge what we read. While this is an excellent practice, we can't get too carried away. In other words, the story also invites us to leave open the possibility of the supernatural, and to recognize that the workings of human (and animal) hearts and minds are infinitely mysterious, no matter how many facts we have under our belts.