One day, Right, a straight-A highschool student, sees a notebook fall from the sky while he is staring out the window in class. After school, he picks it up: it's a black notebook, titled 'Death Note'. It contains instructions on how to use it: write the name of anyone in the book, and within forty seconds, that person will die of a heart attack. But you can also specify the circumstances of the death if you so choose to do so within the deadline of 40 seconds.
Right thinks it's some kind of sick joke. When he sees a report on the TV about a man who has taken some preschoolers hostage, he writes his name in the notebook. And sure enough, after 40 seconds the news comes that the man has suddenly collapsed, dead.
So then Right goes on a secret killing spree, writing the names of noted criminals in the book. He wants to better the world, and eliminate everyone that he doesn't think is 'good'. The shinigami (death spirit) whose Death Note he found comes to visit him. There is no repercussion for using the Death Note (except that, if you used it, you won't be able to enter heaven or hell [my guess: you'll become a shinigami too]). The shinigami is amused at Right's antics, and he decides to stick around to see how things develop.
It's a pretty series with a dark undertone. The direction is pretty intense as well -- they manage to make writing in a notebook look exciting.