To begin with: Death Note is one of my favorite...if not my favorite anime. It has a unique plot line, and you have to keep up with the story or you'll be lost.
Story:
A note belonging to a Shinigami falls to the human world. Anyone whose name is written in the note will die. A student named Light Yagami picks up this notebook, and uses it to rid the world of all criminals.
...Basically, if you yourself found a Death Note, would you use it for the same goal? What would you do if you found a weapon this deadly? Murder is still murder, isn't it?
Art:
I've never seen an anime that has art like this (I'm pretty sure there's some out there, but I never usually watch this genre). It matches beautifully.
Character:
Very well developed characters. Though, of course, they have to be for the idea to work. Logical thinking is a must, and most of the episodes are taken up with what is going through their minds.
Overall, it's an amazing anime. It's the best I've found so far ^ ^