Since this question gets asked a lot... this is my sort of standard response ;)
For a visual response to your question: http://oi44.tinypic.com/20r5nd3.jpg Note that the red zones don't mean "no stay", they just mean (to me) that are not convenient for a short stay in the city, either too far or too rowdy or a bit seedy, but by all means one has to visit them as all have their own charm. Keep in mind though that this is a very personal response, other members will surely recommend different things and probably equally good (or better!)
Just for info I include links to three videos showing the areas of l'Eixample, Gràcia and El Born -as per the above map- so you can have a glimpse on what they look like:
EL BORN: http://infocatalonia.eu/w/xTEtn
L'Eixample is classy and elegant, a large district with wide streets and avenues
El Born is historical with little streets and alleys, may get a bit rowdy at times
Gràcia is like a homy little town within the city with a life of its own, slightly farther (15' by metro) but worth considering.