The Broddring Kingdom was the name of the kingdom of Humans in Alagaësia prior to Galbatorix's uprising and the downfall of the Dragon Riders. However, when Galbatorix slew the last true king of the Broddring Kingdom, he took the title for himself and The Broddring Kingdom then became known as "The Empire" or the Broddring Empire (which the Broddring Kingdom had been expanded into).
Technically, the Broddring Kingdom still existed at the time of the Rider War, though it was "little more than a name on royal decrees" and the majority of the people within the Broddring Empire were unaware of it.
While Galbatorix was called "King", he didn't inherit his title and wasn't intending to give it to his heirs, even though he did not have any: therefore the Broddring Empire - formerly the Broddring Kingdom - was a dictatorship.
The Broddring Kingdom (Empire) was full of poverty and poor quality of life, as well as once children turned eighteen years old, they could be drafted into Galbatorix's army to serve the Empire. Little is known about the Broddring Kingdom before Galbatorix had taken over, though much of the older citizens of the now Broddring Empire believe that it was a much better life in the Broddring Kingdom.