Ogres were a race mentioned only once, in The Hobbit.
Ogres were mentioned in the context of the Riddle Game between Bilbo Baggins and Gollum. After Gollum asked a riddle about what "ruins kingdoms" and "beats tall mountains down", Bilbo tried to think "of all the giants and ogres that he had heard of." None of them, however, were destructive enough to meet the description of the riddle, as the answer was time itself.
It is possible that ogres were a mythical monster in Hobbit folklore rather than an actual race. It is also possible that, like Giants and the term "Fairy", they were simply dropped as Tolkien expanded Middle-earth and the related cosmology.
Other versions of the legendarium
In the earlier legends of Middle-earth written before the release of The Hobbit, ogres were one of the Úvanimor: the monsters, giants, and ogres (and later Wicked dwarves) who were bred from earth by Melkor. They feared the sinews of Tulkas arm and his iron-clad fists, when he had cause for wrath. It is said that Túvo protected Men and Elves from them and from ‘evil fays’, and the Orcs besides. Among the ogres were the Sarqindi, the Cannibal-ogres.
Gothmog was a son of Melkor and the ogress Fuithluin and his name was Strife-and-hatred, and he was Captain of the Balrogs and lord of Melkor's hosts ere fair Ecthelion slew him at the taking of Gondolin.
Portrayal in adaptations
The Hobbit film trilogy
Ogres appear briefly in The Hobbit: The Battle of the Five Armies. They are stated in to have been based off Half Trolls, suggested to be somewhere between the largest orcs and the smallest trolls. They served in Azog's Army as both shock troops and Troll commanders. One commanded a team of catapult trolls. Several Ogres appear in Dale and two are killed by Bard the Bowman. One of the them, notably, had an abnormally small left arm, while his right arm that held his club was more proportionate to the size of his body.
In the extended edition, Azog orders a group of Ogres to destroy the Dwarvish war chariots at the beginning of the battle. The Ogres successfully accomplish their mission, knocking down the machines and killing Dwarves with their maces. Some time later, Azog sends another group of Ogres to bring down the chariot used by Dwalin, Balin, Fili and Kili on their way to Ravenhill. However, the spiked wheels of the chariot behead all the attacking Ogres.