Due to the unstable nature of Elsa's powers, until Elsa could learn to control her winter magic, the castle was closed off for 13 years following the late king's orders; this was done for Elsa's safety, as well as the safety of Anna.
On the day of Elsa's coronation, when her magical abilities were unintentionally exposed to the public by Anna, Elsa accidentally sets off an eternal winter upon running away that froze the kingdom. While Anna searched for Elsa who had traveled into the mountains, Hans took on the task of protecting the kingdom.
Towards the end of the film, thanks to Anna's efforts and love for her sister, Elsa found the courage to undo the eternal winter and peace was restored to the kingdom; Hans was sent back to his homeland to face punishment from his twelve older brothers for his actions of treason against the princess and queen and expelling the Duke of Weselton (a.k.a "Weasel-town") from the kingdom.
Time period
The time period for Frozen is set around the 1840-1850's.
Official crest
The official crest of Arendelle is the crocus, appearing on various items within the kingdom such as banners, capes, flags, uniforms, wallpaper, clothing and jewellery. The colors green and purple, and to a lesser extent yellow, are associated and prominent with the kingdom of Arendelle. Crest designs usually encompass the golden yellow stylized crocus, emblazoned on a background that is purple on the left half and green on the right half. On Elsa's profile banner, the purple left half of the background is darker, and the right half is a darker yellow.
The North Mountain: The highest mountain in Arendelle, on which Elsa sculpted her ice palace. Anna, Kristoff, Olaf and Sven journeyed here on their quest to find Elsa.
The forest: The forest outside of the village that is inhabited by wolves and other animals. It is also where Wandering Oaken's Trading Post and Sauna can be found. Anna journeyed into the forest to find Elsa, only to lose her horse and end up at Wandering Oaken's Trading Post and Sauna and meet Kristoff.
The village: The village outside the castle gates. It is where the Arendelle Citizens live.
Trivia
The name 'Arendelle' is based on the Norwegian town of Arendal, located in the county of Aust-Agder, to the southwest of the Norwegian capital, Oslo. However, the scenery of Arendelle is based primarily on Nærøyfjord in western Norway, as well as various buildings in Oslo, Bergen, and other Norwegian cities.
In Norwegian, 'Aren' is possibly derived from 'ørn', which means "eagle", and 'delle' is derived from 'dal', which means "valley".
The crocus is a symbol of rebirth and spring. In cold regions, the flower blooms while snow is still on the ground.
The eternal winter Elsa caused may be a representation about how in the northern reaches of Norway winter will last anywhere from 8-10 months.
Reading further into the meaning of this kingdom's name, it's interesting to note that Walt Disney's own great-grandfather was named Arundel Disney.
In the film Big Hero 6, a ship from Arendelle can be seen sitting in the docks.
It's unknown who ruled Arendelle until Elsa finally become the queen, seeing as how she didn't have her coronation ceremony until three years after her and Anna's parents died.
Gallery
The Disney Wiki has a collection of images and media related to Arendelle.