The basic plot is, obviously, like the original manga. But there are definitely a lot of differences in how they done things. For those who have not read the manga: first, go by the manga NOW, it's brilliant; second, I'll be nice and tell you the basic plot.
Light Yagami [Tatsuya Fujiwara], the genius son of a police officer. He's studying law and feels VERY strongly about justice, but his bored. So he hacks into his dad's computer to see whether criminals are being punished fairly. To his horror, he see loads of criminals avoid punishment in one way or another!
And that's when he finds: the death note. "The human whose name is written in this note shall die.". So light takes matters in to his own hands, killing those who are unfit to live in the world. As criminal upon criminal die of unexplained heart attacks, the world starts work out that this is all the works of some kind of God, or saint, or (on the flip-side) a devil.