Plaza Catalunya - Real city center of Barcelona
For tourists, the real city center of Barcelona ought to be Plaza Catalunya. It is not only the city center of Barcelona but also the transport hub, where the bus and metro networks connect.
Most of the attractions of Barcelona are at walking distance from Plaza Catalunya. When you need to get to any other part of Barcelona, it is easy from Plaza Catalunya. It is also the starting point for sightseeing tour buses as well as for Airport buses.
Plaza Catalunya is always buzzing with tourists and local population. It has many stores, restaurants, coffee shops. I end up recommending Hotel H10 Catalunya Plaza to my guests as it is in the most convenient central location for accommodation.
At booking.com, I got a better selection of hotels when I selected Eixample district and also 'Show areas popular for sightseeing'. Every hotel listing at booking.com comes with a map. Closer the hotel is to Plaza Catalunya, the more you are in the 'real city center of Barcelona'.
With regard to other neighborhoods for accommodation:
This is heart of the Barcelona old town Ciutat Vella, a favorite with backpackers thanks to the abundance of cheap accommodation. I do not recommend Gothic Quarter as an area to stay in Barcelona.
I think there is an another question by you, where you have asked for areas to avoid for accommodation. My colleague, Nuria, has added Gothic Quarter to the list of areas, one can avoid in Barcelona with regard to stay-locations.
Gracia neighborhood retains a small-town atmosphere and does have an authentic feel to it. It is also home to some of the popular traditional bars and restaurants of Barcelona. My favorite pick for accommodation in Gracia is Casa Gracia, an affordable hotel.
Recommended hotels in Barcelona
Central and affordable - Violeta Hostel
Violeta is also called Elle. It has a perfect location, very close to Plaza Catalunya. The rooms are quite neat, with a very nice décor, for a great price that works out to about 30 to 40 euros per person.
Party zone and affordable - Twentytú Hostel.
I recommend this hotel if you like a lot of walking, as it is close to nightclubs in the Marina area. It has a spectacular roof terrace with incredible views of Torre Agbar. It has a very large shared kitchen. It is frankly very attractively priced.
Central with Gaudi views - Nextto
With incredible views of the famous La Pedrera (Casa Mila), this is an average priced hotel in the modernist district of Eixample. The Nextto decor is fabulous, very tasteful. Another highlight is the great buffet.
Mid range central hotel - Hotel Actual Boutique
Also near La Pedrera, this hotel too has a great location. It is a midrange hotel and is right on Passeig de Gracia.