Tolkien developed the Elvish languages of Middle-earth as a related language family with deep internal history.
The two best-known members are Quenya and Sindarin. Quenya was associated especially with the High Elves and drew inspiration from Finnish and other linguistic sources. Sindarin was the everyday Elvish language of many Elves in Beleriand and later Middle-earth, with influences including Welsh.
Tolkien did not invent only vocabulary lists. He built sound systems, grammar, writing traditions, historical change, and relationships between languages. The fictional history of these tongues is part of the history of the peoples who speak them.
Valarin belongs to the speech of the Valar and is not simply another name for Elvish. The Black Speech was created for Mordor, Westron was the Common Speech, and Adûnaic was the language of Númenor. These distinctions make “Elvish languages” the appropriate family-level answer.