I would say the area called "le quartier des Halles". This is a young and vibrant area. A virtually unlimited choice of cafes/bars and restaurants. Also a multitude of shops and Boutiques. It is an area also very much loved by artists.
The George Pompidou Centre is also located there.(It houses The Publique Library, and the Musee D art Modern (modern art museum) It is the largest museum for modern art in Europe, and a centre for music and acoustic research. It is also called Beaubourg, due to its location (area called Beaubourg) and is well worth a visit.
It is one of the oldest area in Paris, it has character & history.
It is very central and is on the train line called RER A. Those are long blue/red and white trains that run both underground and above ground level. I don t know by which airport you will arrive but if it is Charles De Gaulle, you can get the RER A from Charles de Gaulle straight to Chatelet les Halles. It is a long journey (a good 30 to 40 minutes)
It probably is the most lively area in Paris. It is a good mix of old and new, history and modernism.
Enjoy! Oh, and how do I know that? I am French!