After watching the movies, I was left wondering: where do the Dwarves come from? Do they all originate from the same place, or do they have different "countries"? Also, are the dwarves we meet all related to each other?
The "God" of middle earth "Iluvatar" did not create them. Aulé, the master of craft and work, created the Seven Fathers of the Dwarves deep beneath an unknown mountain somewhere in Middle-Earth. He did not have the power to give life to his creations so the dwarves were bound to him. All other Dwarves are descended from them.
Iluvatar reprimanded Aulé, and as such Aulé decided to destroy the Dwarves. As they cowered and begged for mercy, Ilúvatar took pity on them and gave them true life.
The Seven Fathers awoke in their places in pairs with their wives, though Durin 1 had awoken alone. The seven different groups of Dwarf-folk originated in the locations where the Seven Fathers of the Dwarves first awoke. 3
I hope this answer your question but for more information I suggest you consult this site, http://tolkiengateway.net/wiki/Main_Page
As far as the second part of your question is concerned:
Thorin, son of Thrain, son of Thror, was the heir to the Kingdom under the Mountain. Fili and Kili were his nephews though Dis, his sister (who, incidentally, is the only female Dwarf named in all Tolkien's works).
Balin and Dwalin were brothers, as were Oin and Gloin. All four were Thorin's third cousins.
Dori, Nori and Ori were also brothers, but were only distantly related to Thorin.
Bifur, Bofur and Bombur were cousins and were descended from the Dwarves of Moria.
Dain, son of Nain, who comes to the Battle of Five Armies with five hundred Dwarves, is the second cousin of Thorin and became King under the Mountain after Thorin's death. According to the books, Dain was still the King during the time of the War of the Ring, but fell defending the body of King Brand (Bard's grandson) when Dale was attacked and the Mountain laid siege to by Sauron's forces at the same time he was attacking Gondor.