OK, this is how we're memorizing the dwarf names at the Potts House. The dwarves can be organized into five groups of relatives: two groups of two, and three groups of three. Each pair is two brothers, and each trio is two brothers plus another brother, cousin, or uncle.
You can memorize the nickname for each group and rattle them off using the fingers of one hand. The groups are the ALINS, the ÓINS, the ILIS, the ORIS, and the URS. (They run almost like the list of vowels: A, O, I, O, U). Remember this list, and "two twos, three threes," and you're almost there.
The 2 ALINS ("ah-linz") are the brothers BALIN and DWALIN.
The 2 ÓINS ("oh-inz") are the brothers ÓIN and GLÓIN.
The 3 ILIS ("ee-leez") are the brothers FILI and KILI and their uncle THORIN.
The 3 ORIS ("or-eez") are the brothers ORI, NORI, and DORI.
The 3 URS ("urz") are the brothers BOFUR and BOMBUR and their cousin BIFUR.
Got that? Practice counting the five groups off on the fingers of one hand and reciting the members of each group a few times, and you'll be a true Tolkien nerd!
There are two naming rules that apply to most of the the thirteen dwarves.
First, the "brother name" rule: if the names differ only by opening (or missing) consonant sounds, they are brothers: Balin and Dwalin; Glóin and Óin; Fili and Kili; Dori, Nori, and Ori. Bombur's name breaks this rule. He is Bofur's brother. If his name followed the rule, it would have been something like Grofur or Ofur.
Second, the "cousin name" rule: if dwarves have the same last syllable but differ by a vowel sound before that syllable, they are cousins. So: brothers Balin and Dwalin are the cousins of brothers Glóin and Óin; Bifur is the cousin of Bofur.
I don't have an rule for Thorin. His name makes it sound like he could be related to Glóin and Óin, but he's not. You'll just have to remember that he goes with Fili and Kili.
Now for a few details to help you get their biographies and faces into your head. This will help you recognize them easily in the movies and know just a bit about their roles in the group.
Balin has a big white ski-slope beard and winning smile. He's the second dwarf that shows up at Bilbo's hole. He will later (after the events of _The Hobbit_) lead an expedition to retake Moria from the orcs, and wind up in the tomb there, eventually discovered by Gimli.
Dwalin is the only dwarf with a bald head, and his scalp is covered with tattoos. He's the younger brother of Balin, and the first dwarf that shows up at Bilbo's hole.
Óin is an older dwarf, with coiled braids in his beard, a gray mustache, and knit gloves. He is somewhat deaf, and uses an ear trumpet. He will eventually be killed by the "Watcher in the Water" at the gate of Moria.
Glóin is the father of Gimli, and looks an older, red-haired version of Gimli. To remember which one is Gimli's father, keep in mind that Gimli and Glóin both start with G, and Glóin is the only dwarf in the group whose name starts with G. In the book, Glóin and Óin are responsible for kindling "glowing" fires.
Note: Balin and Dwalin are first cousins to Óin and Glóin. Their father, Fundin, is the brother of Gróin, the father of Óin and Glóin. We could have put the four of them into the same group, the INS, but it seemed easier to memorize the dwarves primarily as groups of brothers, with two exceptions, rather than mixing up the groups. If you prefer, think of the ALINS and the ÓINS as one big group and use the brother and cousin naming rules to keep track of their relationships.
Fili has hair that is dirty blonde, and little beads in his braided mustache. Fili and Kili are the youngest of the group. These two brothers and Thorin all have smaller noses and more human-looking features than the other dwarves.
Kili has long, dark brown hair that is not braided. He has smoldering dark eyes and only a very minimal beard and mustache (perhaps he's just out of Dwarf puberty). Tolkien's own writings differ on whether Fili or Kili was youngest; Appendix A of _Lord of the Rings_ says Kili is youngest, and it looks like the film producers went with that interpretation.
Thorin Oakenshield is the party's leader, king in exile, and the uncle of Fili and Kili. He looks quite a bit like Kili, but his hair is just starting to go gray, and has a more regal bearing. In the movie he wears a big fur coat and uses an oak log as a shield.
Ori is one dorky-looking dwarf. He has the bowl-cut bangs and an assortment of braids that make him look a bit like a Hasidic Jew, and a wispy mustache. He's the scribe of the group, and often seen writing. He will eventually write the last words in the Book of Mazarbul, in Balin's chamber in Moria, which say "We cannot get out. A shadow moves in the dark. We cannot get out. They are coming." Poor Ori!
Nori continues his brother Ori's Hasidic theme with a six-pointed, star of David-shaped hairdo, complete with metal beads everywhere, even on his braided eyebrows. He definitely wins the "most ridiculous dwarf hair" contest.
Dori is the oldest of these three brothers and has silver hair, bushy eyebrows, a long mustache and strikingly light-colored gray eyes. He also has lots of braids and beads, and metal ear cuffs.
Bofur has the winged hat and an impressive Salvador Dali-style curled mustache to match. He's one of the friendlier dwarves. Like Balin, he's usually smiling, and he has a touching scene with Bilbo in the Misty Mountains, right before the party is captured by orcs.
Bombur is the "bomber" -- he's the biggest dwarf, with the red beard that forms a loop, and fingerless gloves. He breaks the furniture at Rivendell due to his weight. In Tolkien's books we learn that he will eventually require six younger dwarves to carry him.
Note: the two brothers in this group have names that are more similar (both start with "Bo"). Bombur's name breaks the brother name rule.
Bifur is the cousin of Bofur and Bombur. He the one with the bifur-cated head (sort of) -- he's got an orc's axe blade stuck in his forehead. He's also got a bi-color beard -- black and gray braided together. His black hair is unkempt, his uneven eyes make him look slightly crazy, and he only speaks Dwarvish. This probably has something to do with his little "accident."