Thorin and Company were a group of fifteen, thirteen dwarves, one hobbit and one wizard. Their names were Thorin Oakenshield (the most important of the dwarves), Balin, Dwalin, Fili, Kili, Dori, Ori, Nori, Oin, Gloin, Bifur, Bofur, Bombur, Bilbo Baggins (the hobbit, and burglar for the company), and Gandalf (the wizard).
The above went on an expedition, planned by Thorin II, to recover their gold from the Dragon Smaug in Erebor. They all got to the mountain together, but Gandalf would go off on "important business" from time to time.
The gold was taken back and Erebor restored, however more important things happened during this quest, some which don't become apparent until The Lord of the Rings. Bilbo found the One Ring, Gandalf found the sword Glamdring, and slew the Great Goblin of the Misty Mountains, Smaug (the greatest evil in the land) was slain, and the Battle of The Fives Armies took place, in which Thorin, Fili and Kili were all slain.