It means the Gothic Quarter, actually meaning the Old City. And if it's your first time around Barcelona, it's probably the first place you'll wind up. It is also most likely that your hotel or apartment is somewhere here, as this part of the city is the # 1 tourist attraction whatever travel guide you consult.
Barri Gothic possesses all that's inherent to a typical Catalonian city: Narrow streets, old houses, magnificant structures like cathedrals, museums, parks, forums, public squares and occasional street cafes just around any corner. Not to mention hordes of wandering tourists... Streets are a like a giant spider web. Don't be afraid to get lost. Do get lost.