The Death Note in the title is really a notebook. Any person who's name is written in the notebook will die, when and how it's written. It's normally the property of the shinigami Ryuk, but now it's fallen into the hands of a human named Light. (I have to say, I totally love how Ryuk seems to have let a human get his Death Note just for the lulz.) Light is a brilliant student, so he's able to use the Death Note more efficiently than perhaps anyone else could have. His goal is to eliminate crime by using the Death Note to systematically kill criminals. Naturally, this eventually comes to police attention: dozens of criminals, most of them already in police custody, suddenly drop dead from a heart attack every day. It makes people talk. This leads up to secretive super detective L. sitting down to match his wits with Light, who seems to be up to the task.The deaths are almost all offscreen, and the book is mostly Light talking with Ryuk about his plans, what he's doing currently, and why. He's very, very clever. He uses the abilities of the Note, as he discovers them, to try and through L. off his trail. And he does a good job, for the most part, but there are occassional slip-ups. This is, I assume, the first time he's played the part of a mystical serial killer. L., for his part, seems to be right behind most of the time, picking up on the messages that Light wants to send, and most of all knowing that they're being deliberately sent.
You know how, when characters in movies have invisible companions that nobody else can see, they spend an awful lot of time talking to them where tons of people can see them talking to themselves? I usually dislike this, so I was relieved to see that Light doesn't speak to Ryuk (who nobody else can see) around other people. Because he's too smart for that, thank God.
There's an interesting moral question set up in the first volume, that I imagine will stretch throughout the manga. So what if Light is killing a man who just murdered his entire family? Doesn't he deserve it anyways? But now Light is responsible for the deaths of hundreds, and intends to kill hundreds more. What does he deserve? I never expected this, but I really am eager to read the rest of the series now.