That gate to Moria was created in the "Second Age" of Middle Earth, and was used to trade mithril with the Noldorian Elves of Hithlum. Relations between the Elves and the Dwarves were more cordial in the Second Age. The inscription and password on the gate created by Celebrimbor, the leader of the Elves of Noldor, hence is in elvish. Celebrimbor also forged the 3 Elvish rings of power.
[This is all in the Silmarillion]
In The Fellowship of the Ring, the inscription on the door reads:
Im Narvi hain echant. Celebrimbor o Eregion teithant i thiw hin
"I, Narvi, made them. Celebrimbor of Eregion drew these signs"
(taken from Wikipedia, I hope it is accurate). So Narvi the dwarf built the gate, and Celebrimbor drew the lettering and the cast the spell which only opens when the password is given.