"Moria Goblins" refers to the orcs that resided in and had overtaken Moria.
These goblins were akin to the 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. When Thror came to reclaim Moria with Nar, Azog killed Thror, initiating 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. 
Portrayal in AdaptationsEdit
Peter Jackson's The Fellowship of the RingEdit
In Peter Jackson's first movie, Moria Goblins are shown scuttling out of cracks in the ceiling of Dwarrowdelf like cockroaches. They fight the Fellowship in the caves of Moria with a Cave-troll. They are eventually scared off by Durin's Bane.