The Grandfather of Thorin II Oakenshield, Thror was the King Under the Mountain who discovered the Arkenstone. Driven from Erebor by Smaug, Thror tried to reclaim the lost Dwarf Kingdom of Moria but he was unfortunately killed by Azog.
Thrain, Son of Thror
Thorin II Oakenshield's Father, Thrain led the Dwarves in the great Dwarf and Orc war. After Azog was killed, marking the end of the war, Thrain tried to take back Erebor but was captured and tortured to death by the Necromancer. Before his death, he gave the map and key to Erebor to Gandalf.
These are my interpretations of Thror and Thrain, taking aspects from both the books and films. Hope you enjoy! Comments and criticisms appreciated.