The Bus Turístic (Touristic Bus) is the easiest and best way to get to know Barcelona as it allows to hop on hop off as many times as you want with the same ticket allowing you to visit the main attractions of Barcelona.
There are 3 different routes that cover the entire city, Roja (RED), Azul (BLUE) and Verde (GREN). The best routes are by far the RED and the BLUE ones, the GREEN one covers and area that it is not specially beautiful or has something interesting to see so I would skip the GREEN one.
The BLUE route includes the Montjuïc area (Olympic Stadium, Poble Espanyol, MNAC, the Joan Miró Foundation, …) and has great views of the city and the Port so those of you that like taking pictures will love it. This route also covers the Port Olímpic, Ciutadella Park (Parliament and the Zoo), Plaça Catalunya and the Passeig the Gracia where it is possible to visit to of the most beautiful buildings of the city, the Pedrera and the Casa Batlló, both designed by the famous architect Antoni Gaudí.
The RED route includes the Plaça Catalunya, the Passeig de Gràcia where it is possible to visit the Pedrera and the Casa Batlló, the Sagrada Familia (designed by Antoni Gaudí) the Park Güell (again designed by Antoni Gaudí), the Monastir de Pedralbes and the Camp Nou (FC Barcelona stadium).
Tip: The GREEN route is quite poor so I would focus only on the RED and BLUE ones.
1 Day adult: €21.00
1 Day child (age 4-12): €13.00
2 Days adult: €27.00
2 Days child (age 4-12): €17.00
Where can tickets be bought?
At any bus, just ask the bus driver.
Tickets can also be bought online. However in my opinion it is not recommended as the price is exactly the same as if you bought the ticket directly to the bus driver (no discount) and you have to pay the posting of the tickets to your home address (or wherever you want them posted to) so it is too much hassle.
* Thanks to Gabriel for the picture.