Gimli, like Legolas, is mainly around to fill out the Fellowship's roster with one Dwarf. Because Gimli is a Dwarf, he is short, bearded, and carries an ax. But he does have a distinct character arc in The Fellowship of the Ring that is worth recording.
Gimli is desperately eager to see the Mines of Moria; he wants to know what has happened to his kinsman Balin and the party of Dwarves who attempted to resettle the Mines thirty years before (hint: nothing good). Not only does he want to know the fate of his family, but he also wants to see the long-lost home of the Dwarves. Imagine if Washington, D.C. had been taken over by a Balrog of Morgoth two hundred years ago. Wouldn't you be eager to see this influential city of the US if you had the chance, even if it were empty for decades?
The problem is, like the Elves, the Dwarves are beginning to fade away. They just don't have the numbers that humans (or Hobbits) have, and they are retreating farther and farther into their mountain tunnels. Gimli's wish to visit the Mines of Moria is deeply melancholy, because he wants to glimpse the Dwarves' past glories – since their present glories are much less grand.
As the Fellowship heads towards Caradhras, Gimli murmurs, "Dark is the water of Kheled-zâram [...] and cold are the springs of Kibil-nâla. My heart trembles at the thought that I may see them soon" (2.3.98). Yet, despite Gimli's pleasure at the thought of seeing the ancient halls of his ancestors, his joy is ruined by the swarms of Orcs, the cave trolls, and the Balrog that pulls Gandalf into the pit beneath the Bridge of Khazad-dûm. The Dwarves' old home has been permanently lost. Gimli has gotten glimpses of the Mirrormere (Kheled-zâram) and Dwarrowdelf, but he has not really recaptured that lost Dwarf greatness.
There is one ray of sunshine in the gloom that is Gimli's visit to the Mines of Moria, and that is his meeting with Galadriel. Gimli and Legolas (and all the other Elves) have not gotten along all that well up until now, since their people do not trust each other (and then there's the matter of Legolas' father imprisoning Gimli's father in The Hobbit: embarrassing!).
But once Gimli arrives in Lothlórien, heartsick over the losses of Moria and Gandalf, he meets Galadriel, and her sympathy for Gimli immediately strikes a chord with him. Gimli falls quickly for Galadriel. When they part, as Gimli continues on his quest, he asks only for a single strand of her hair that he might place in crystal as a treasure for his household. Galadriel is touched (and a bit shocked) by Gimli's daring, but she agrees. Galadriel's compassion so moves Gimli's heart that he leaves Lothlórien forever changed (and friends with Legolas, to boot). So Gimli has shown himself lucky to be the first Dwarf to see the Naith of Lothlórien, the deepest part of this Elf kingdom, since Durin's day.