La Rambla ...
... is the street from the Placa de Catlunya down to the Port Vell, there's a broad pedestrian area in the middle and narrow traffic lines on the sides. This street is lined with cafes and restaurants an basically crowded all day and beyond midnight.
At the end of La Rambla (at the harbour), there's the Columbus monument, built in 1880s. You can take the lift up to the "feet of Columbus" and have a good overview from there over the harbour and the city.
Port Vell ...
... in the 1980s the old port of the city was transformed to a fun place to hang out. Today you'll find there a big mall, restaurants, IMAX movie theatre and the big aquarium.