Plaça Sant Jaume is the center of power of the city. You will see that there are two large buildings: the city hall on the side closest to home, and the regional government headquarters on the other side. One way to distinguish them is by the flags, since the City Hall has the one of Barcelona.
You will see that in Barcelona there are many flags and symbols not only in official buildings but also in many balconies. Here is a short summary of some of them. You can also read this other post about the political conflict in Catalonia.
But we were in the Plaza Sant Jaume. Actually the square is from the 19th century since there was a church before, as in most large spaces of the old city. In the 19th century many religious buildings were demolished and public places and facilities were opened, as we explain for example in this post.
The two facades are modern. If you want to see the medieval façades, you have to look for the Carrer Ciutat for the City Hall, and the Carrer Sant Honorat for the government of Catalonia.
The most curious thing is that this area has been the center of power since the foundation of Barcelona. In Roman times the forum (the public square) of the city was approximately in the same place, with the temple of Augustus. Now there are only 4 columns of the temple, which you can see in the patio of the house of Carrer Paradís 10 (look at the schedules). Not many people know this place, it is curious.
If you look, right here is the highest point of what was the mountain Taber, which is where Barcelona was founded. Actually if you look every street do rise to Plaça Sant Jaume.