Beside Frodo who volunteered to be the Ring bearer and his Hobbits friends who volunteered as well, the rest of the Fellowship members were named or chosen by Elrond:

‘And I will choose you companions to go with you, as far as they will or fortune allows. The number must be few, since your hope is in speed and secrecy. Had I a host of Elves in armour of the Elder Days, it would avail little, save to arouse the power of Mordor.

‘The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil. With you and your faithful servant, Gandalf will go; for this shall be his great task, and maybe the end of his labours.

‘For the rest, they shall represent the other Free Peoples of the World: Elves, Dwarves, and Men. Legolas shall be for the Elves; and Gimli son of Glóin for the Dwarves. They are willing to go at least to the passes of the Mountains, and maybe beyond. For men you shall have Aragorn son of Arathorn, for the Ring of Isildur concerns him closely.’

The Lord of the Rings Book Two, Chapter 3: The Ring Goes South Page 275-6 (Single volume 50th Anniversary Edition)

The question is why Legolas and Gimli were chosen and not other representatives of their races?

Possible answers:

1. Gimli and Gloin were the only dwarves in Rivendell that time, and Gloin was too old for the quest.
2. Gimli and Legolas were among the most formidable warriors of their races, and thus Elrond chosen the best fighters to escort and protect Frodo.

Beside Frodo who volunteered to be the Ring bearer and his Hobbits friends who volunteered as well, the rest of the Fellowship members were named or chosen by Elrond:

‘And I will choose you companions to go with you, as far as they will or fortune allows. The number must be few, since your hope is in speed and secrecy. Had I a host of Elves in armour of the Elder Days, it would avail little, save to arouse the power of Mordor.

‘The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil. With you and your faithful servant, Gandalf will go; for this shall be his great task, and maybe the end of his labours.

‘For the rest, they shall represent the other Free Peoples of the World: Elves, Dwarves, and Men. Legolas shall be for the Elves; and Gimli son of Glóin for the Dwarves. They are willing to go at least to the passes of the Mountains, and maybe beyond. For men you shall have Aragorn son of Arathorn, for the Ring of Isildur concerns him closely.’

The Lord of the Rings Book Two, Chapter 3: The Ring Goes South Page 275-6 (Single volume 50th Anniversary Edition)

The question is why Legolas and Gimli were chosen and not other representatives of their races.

Possible answers:

1. Gimli and Gloin were the only dwarves in Rivendell at that time, and Gloin was too old for the quest.
2. Gimli and Legolas were among the most formidable warriors of their races, and thus Elrond has chosen the best fighters to escort and protect Frodo.