Elvish

Elvish may refer to any language spoken by Elves, especially either of the two major Elven languages in J.R.R. Tolkien's legendarium, Sindarin and Quenya.

It is not the name of any Elven language in Arda, and particularly nitpicky agents may get tetchy if it's used as such. Particularly if the language being used is actually Grelvish.

Elven Languages of Middle-earth
Sindarin is the most commonly spoken Elven language in the Third Age, when The Lord of the Rings takes place. Historically, it is the language of the Sindar (or Grey Elves), who began the journey to Valinor but ultimately remained in Middle-earth. Third-Age people speaking of "Elvish" or the "Elven-tongue" usually mean Sindarin.

Quenya is the language that was spoken by the Elves who made the journey to Valinor. When the Noldor returned to Middle-earth in pursuit of Morgoth and the Silmarils, they brought Noldorin Quenya with them. They adopted the local Sindarin as their vernacular, but continued to use Quenya among themselves. By the Third Age, Quenya is to the Common Tongue (Westron) as Latin is to modern English, and its usage is mainly scholarly or ceremonial, and is referred to as the "Ancient Speech" or "High Speech."

The Elven languages may be written using either Tengwar or Cirth. Tengwar is the curving script most people picture when they think of written Elvish. Cirth is the runic script used, for example, to inscribe the Doors of Moria.