The story begins in a Japanese high school where the highly intelligent (and thereby succesful) student Light Yagami finds a notebook, which a Shinigami (Japanese god of dead) dropped there intentionally out of boredom. The owner of the notebook named "Death Note" does have the ability to kill people by writing their name into it while imagining their faces.There is a certain ruleset which determines the patterns by which the notebook can be used. Those rules are revealed eventually throughout the series and shall not be introduced here any further.
The Shinigami named Ryuk soon realizes, that having Yagami finding the notebook, turns out to be a promising cirumstance according to the entertainment he is looking for, because great plans are taking shape in Light Yagamis mind:
Yagamis view of justice is based upon the believe that there are crimes, which make those who commit them, lose their right to live. Therefore he begins killing criminals of the highest category using the death note. However, as the suspicious incidents of dying imprisoned criminals rise above normal, people begin to scrutinize, about someone being responsible for the deaths. One of them is "L".
"L" is a mysterious detective of whom nothing more is known, than that he is a genius solving complicated cases already in the past.
Not even little less intelligent than Light, a great story of chess-like trap-laying, -avoiding and bait taking unfolds around the two given individuals. Their colliding understandings of justice split society into half and even if the manga-author denies any philosophical intention of the story I personally cannot think of a story that illustrates the conflict of two contradictionary views on a subject like (in this case) justice and ethics better than "Death Note". At least in the first half of the series. Later the conflict and the "chess-game-like" character of the rivalry does fade out a little, but the first half of the series makes up for the weaknesses of the latter half.
To round this outline I'd like to mention a view technical details on this anime. The soundtrack places itself somewhere between rock and metal and is not the worst I ever heard in anime. A big portion of gothic-loli serves for the fan-service-faction of otakus and the quality of the drawings is upper standard nowadays.
To conclude the review I want to say that "Death Note" is a duty for everyone who enjoys a twisted intensive story, a philosophical discussion or chess. Since this anime is often considered silimar to Code:Geass I want to add, that indeed there are paraleles, as there is also a great manipulating main character, doing morally questionable things, to make the world better. For this aspect Code:Geass does have the upper hand. But in storytelling and -twisting and unfolding traps, Death Note sets the bar very high and is still uncontested.