What is the name of the language of the Orcs? Was it developed by Tolkien as much as Quenya?
One of their languages is (drum roll...) orkish
Orkish was a general term for the jumble of languages used by the Orcs, composed from corrupted borrowings from other languages of Middle-earth. The variations in Orkish between different tribes and types of Orc were so great that it was often useless for communication, and so a standard language was necessary. Sauron devised his Black Speech for this purpose in Mordor, but in practice the Common Tongue was more normally used.
As you see, they also spoke the Black Speech, which was the language among all the servants of Mordor, thus including orcs. It was created by Sauron in order to make the orcs and the rest of his servants understand each other. When Sauron was overthrown (during the Second Age), the ancient "pure" form was forgotten by all but the Nazgûl.
The language that was used for inscription upon the One Ring was 'pure' Black speech (J.R.R. Tolkien, The Lord of the Rings, The Fellowship of the Ring, "The Council of Elrond")
The Black Speech was at least in part based on Valarin. The Black Speech was incomplete and there weren't any poems or songs written in it. Tolkien didn't develop it as much as the other languages.
The Black Speech is one of the more fragmentary languages in the novels. Unlike Elvish, Tolkien did not write songs or poems in the Black Speech, apart from the One Ring inscription.
The Black Speech was not intentionally modeled on any style, but was meant to be self consistent, very different from Elvish, yet organized and expressive, as would be expected of a device of Sauron before his complete corruption. It was evidently an agglutinative language. [...] I have tried to play fair linguistically, and it is meant to have a meaning not be a mere casual group of nasty noises, though an accurate transcription would even nowadays only be printable in the higher and artistically more advanced form of literature. According to my taste such things are best left to Orcs, ancient and modern.
(J.R.R. Tolkien, "Words, Phrases and Passages in Various Tongues in The Lord of the Rings", Parma Eldalemberon 17, p. 11-12.)
To complement @Shevilaskovic's answer: Common was also used, because orcs of different tribes could not understand each other, as can be seen in the passage when Pippin and Merry were captured by orcs:
To Pippin's surprise he found that much of the talk was intelligible: many of the Orcs were using ordinary language. Apparently the members of two or three quite different tribes were present, and they could not understand one another's orc-speech. There was an angry debate concerning what they were to do now: which way they were to take and what should be done with the prisoners.