"Moria Orcs" refers to the orcs who in TA 2480  came to inhabit the Dwarven realm of Khazad-dum, at the time Sauron populated "Moria with his creatures", long after the Dwarves had fled there in 1981.
Three centuries later, Azog was their leader. When Thror came to reclaim Moria with Nar, Azog slew Thror, and this initiated The War of Dwarves and Orcs. The Dwarves hunted down Azog until he was killed by Dain II Ironfoot at the Battle of Azanulbizar. (After Azog's death, his son Bolg would lead an army of Orcs to Erebor, where they lost terribly, and Bolg was killed.) When the Dwarves, led by Balin, returned to Moria in TA 2989, they came upon bands of orcs who eventually invaded and overtook Khazad-dum, and Balin was slain. These orcs were not large in number or well-armed, but sufficient to keep out any enemies.
The Fellowship of the Ring came upon a large group of Moria Orcs, led by an unknown large orc chieftain, who ambushed them in Balin's Tomb. After the Fellowship left the chamber they came upon other orcs and Cave-trolls who attempted to kill them as they neared the Bridge of Khazad-dûm. The Moria Orcs stopped due to the arrival of Durin's Bane, the ancient Balrog of the old kingdom. After the Balrog's defeat, Moria Orcs continued their hunt for the Fellowship to Lothlórien where they were killed by the Elves living there.
While most Moria Orcs served Sauron, some of them fell under the influence of the wizard Saruman or acted on their own behalf. Several of the Moria Orcs in the service of Saruman along fellow Mordor Orcs and Uruk-hai were sent by the old Istari to capture Frodo Baggins and get the One Ring. The orcs however took Meriadoc Brandybuck and Peregrin Took in place of Baggins and began their journey back to Isengard. At some point during this journey, the orcs of the Black Pit grew hungry and began arguing with Uglúk about eating the two hobbits. Uglúk had to prevent any riot so he killed several of the traitorous orcs and warned the others. The surviving Moria Orcs were killed either on the road by the Riders of Rohan or at night near Fangorn Forest by Eomer's Rohirrim group.
After the destruction of Isengard, these orcs were driven out of Moria and the Dwarves reclaimed it.
Portrayal in adaptations Edit
Peter Jackson's The Hobbit Trilogy Edit
In The Hobbit: An Unexpected Journey, legions of orcs are being led by the Gundabad Orc known as Azog into Moria. These orcs fight against the Dwarves led by Thorin Oakenshield, his father Thrain and his grandfather Thror. Azog manages to slay King Thror, but loses an arm to Thorin, who then leads his dwarves against the remaining orcs, while Azog is taken by his guards back into Moria. Later, Azog is revealed to be the leader of a hunter orc band that is attempting to kill the company of Thorin that seeks to reach Erebor and kill Smaug. (The orcs that fought at the Gate of Azanulbizar were the first to be called Moria Orcs, however it's unknown from where they came though it is likely that like Azog they were from Gundabad.)
In The Hobbit: The Desolation of Smaug, the Necromancer, secretly the Dark Lord Sauron who is regaining his strength, is revealed by Beorn to have made a pact with Azog, in which the orc huntsman will rally the orcs of the Misty Mountains to their cause. Later in the film, Moria orcs are being summoned by Sauron to his fortress of Dol Guldur and puts Azog in command of the newly formed Guldur army and will lead them to Erebor in order to stop the dwarves from reaching the mountain. Near the end of the film, the armies of Moria Orcs and Wargs are seen leaving Dol Guldur.
Peter Jackson's The Fellowship of the Ring Edit
In The Fellowship of the Ring, after Pippin accidentally knocks a bucket into the dark crevasses and cess pits of Moria, sounds of drums and orcs are heard and the Fellowship of the Ring lock themselves in Balin's Tomb. The doors are destroyed and bands of Moria Orcs (smaller than the ones in The Hobbit Film Trilogy) attack them. The orcs and Cave-troll are dispatched of and the Fellowship led by Gandalf enters one of the great halls with columns, only to see orcs on climbing down from crevasses and walls. These orcs circle them, but are quickly scared by their terrifying ally, Durin's Bane. Later other orcs shoot at them with arrows, while the Fellowship was fleeing from the Balrog.