"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.)
Later many orcs sent themselves from Isengard to invade and take over Moria again. When the Dwarves, led by Balin, returned to Moria in TA 2989, the orcs eventually invaded and overtook Khazad-dum, and Balin was slain.
After the destruction of Isengard, these orcs were driven out of Moria and the Dwarves reclaimed it.