The Moria Goblins is the term given to goblins that resided in Moria.
Moria Goblins (the Goblins that dwelt in Moria) were similar to other Orcs of the Misty Mountains. After the Balrog killed King Durin VI, Goblin bands began to raid Moria, pillaging it of its gold, jewels, and Mithril. They were lead by Azog, however, Thror came to reclaim Moria with Nar. Azog killed Thror, starting The War of Dwarves and Orcs. The dwarves hunted down Azog until he was killed by Dain II Ironfoot in The Battle of Azanulbizar. After Azog's death his son Bolg led an army of Orcs (possibly including ones from Moria) to The Battle of Five Armies, where they lost terribly and Bolg was slain. Later when Sauron started gathering his forces he sent many goblins and cave trolls to Moria, where they multiplied. When the Dwarves, led by Balin, returned to Moria, the Goblins attacked and killed them, though many were slain. More were killed when the Fellowship passed through Moria, and still more when they pursued their quarry to Lothlórien and were killed by Elves there. In battle, these Goblins used swords, bows, and knives. One "huge Goblin-chieftain, almost man-high, clad in black mail from head to foot," carried a great spear (with which he struck Frodo) and a scimitar.  Later on, a party of Moria Orcs joined forces with the Uruk-hai Scouts of Uglúk (and the Orcs under Grishnákh) since they wanted to get revenge on The Fellowship, but most of the ones that wanted to kill the Hobbits were slain by Ugluk, and the remainder were slain by Eomer's Rohirrim 
Portrayal in AdaptationsEdit
Peter Jackson's The Fellowship of the RingEdit
In Peter Jackson's movie, Moria Goblins are shown scuttling out of cracks in the ceiling like cockroaches. They later fight the Fellowship in the caves of Moria with a Cave-troll. They are eventually scared off by Durin's Bane.