University
The university has many interesting buildings, libraries and courtyards.
harbor
Although it's not a quarter, can be clearly separated from a visitor's perspective.
To find it, just walk down La Rambla until the monument to Columbus. The sea is just after it. The harbor you can see is completely new, since until the 80's it was closed to the public. Some old buildings related to the harbor still exist, such as the Tinglado near the Barceloneta quarter that is now a history museum.
There are two main parts in the harbor: the Moll de la Fusta, which are the decks that touch the city, and the Maremagnum, a newly created amusement center/shopping mall, with an Aquarium, an IMAX cinema, discos, bars, restaurants. The clubs are situated on the top floor, which is accessed through the general shopping area via stairs and the like. Even out of season and on work nights you are sure to have a good time, as the clubs are always filled.
It is located at the other side of the wooden bridge next to the Columbus monument.
Guell Palace
While it is on renovation, with only basement and ground floor open for visitors, it is worth visit only if there is no queue to enter.