One of two sub-branches of Celtic languages, also known as "Goidelic". It now consists of Scots and Irish Gaelic. Manx, a now extinct language akin to Gaelic, was also Goidelic. It is believed by some linguists and historians that the language or languages of the Celtiberians, Celtic peoples living in what is now Spain, were also Goidelic. (As a side note, one of the ancient names for Celtiberian Spain was "Galidicia", which may be akin to "Gaul", the name of Celtic France, and may also be the source of the word "Gael", "Gaelic", etc.)
The designation "Q-Celtic" arose because one Indo-European sound tends to manifest itself in the Goidelic tongues as a "Q" (/k/ or /kw/) sound. One example is the Gaelic "mac", "son of", which ends in a /k/ sound; the Welsh equivalent, "ap", "son of", obviously ends in a /p/ sound.
See also P-Celtic.