At the very center of Barcelona lies the Gothic Quarter, or Barri Gotic. It stretches all the way from the waterfront to the Via Laietana, and as it is neighbouring the busy Ramblas it is most likely the first point of contact you will have when visiting Catalonia's capital.
Most buildings here date from Medieval times, some of them go as far back as Roman settlements of the area. This is reflected in the layout of the quarters, which is literally a labyrinth of narrow and convoluted streets, and it can take less than 5 minutes to feel disorientated.
But that is not necessarily a bad thing; my advice is actually to visit the quarters completely unprepared and without a map. It is quite a sensation to feel lost in this medieval backdrop, unexpected discoveries waiting at every corner.
This is also a great place for shopping, eating, drinking and going out, as the quarters are brimming with life at pretty much any time of day (and night). Night walks after a nice dinner and few drinks can be especially rewarding as you will experience the city from a whole different perspective, staged and illuminated by the dim, ambient street lights.
Be careful however, Barcelona is pickpocketing central and this applies especially to the Ramblas and the Gothic Quarters. This was my 5 visit here, and to my shame I have to admit that this was the first time I haven't been pick-pocketed or mugged. Never leave anything on the table in front of you and only take as much money as you need.
Also you might want to bypass the Ramblas after 2 am unless you are ready to deal with all kinds of underworld characters, which by the way can also be quite an experience.