Fili and Kili are brothers; they're also the two youngest dwarves in the group. In general, because they are younger, they seem more energetic and eager to explore than the others. They are responsible for finding the cave in the Misty Mountains where the company shelters during a storm (only to be found by goblins in the middle of the night).
They also seem to be less totally in love with gold than the rest of their countrymen. When they all pile in to the central chamber with Smaug's gold, Fili and Kili grab two golden harps off the walls and start to play music rather than pick through the treasure looking for things they want. Fili and Kili also seem uncomfortable but willing to accompany Bilbo down into the dragon's passage, unlike any of the other dwarves (12.5).
Sadly, Fili and Kili die defending Thorin from the goblins during Thorin's last stand. They are particularly loyal to Thorin because he's their uncle. We have seen the importance of family loyalty in dwarf culture in other places – for example, when Dain of the Iron Hills comes to save Thorin when Bard and the Elvenking bring armies to the Lonely Mountain. So we guess that Fili and Kili would be proud to know that they gave their lives for the sake of family.