It's about a boring, top-notch honor student named Light who finds a notebook which belonged to a death god. He then opens it and finds out about the rules it has:
Basically, if you write a name inside of it, the person dies in 40 seconds. But only if the person who writes the name in it knows or has seen a phyiscal description in his mind, this is to prevent the user from killing 5 people with that instance of the name.
And also the type of death needs to be specified, or the person simply dies from heart failure.
He then gets visited by Ryuk, the Death God who dropped the notebook into the human world. And tells him he can use it, but he isn't his ally (as he proclaimed that one day he will write his name in his).
Light then becomes crazed with the book's power and makes a goal: To kill off all the criminals, so humanity can live in an utopia of peace and he rules it as a God.
It's one of those animes that you have to be into to stay into.