The Turtle Dove is smaller than a Collared Dove. Its head is blue-grey with a black and white patch on the side of the neck. The throat and breast are pale pink and the belly white. The wings and back have black and chestnut tortoiseshell-like markings, but this is not why it's called the Turtle Dove.
The "common turtle-dove" (European turtle-dove/Streptopelia turtur/Columba Turtur) has no connection to turtles, but instead, its name was one that it bestowed upon itself: their specific name, "turtur", is a Latinisation of their purring “turr turr turr” song.
In Catholic Christian symbolism, doves and turtle-doves were the only birds that could be offered in sacrifice, as they were clean according to the Mosaic law (Ge. 15:9; Lev. 5:7; 12:6; Luke 2:24).
So for the song "The Twelve Days of Christmas", two turtle doves represent the Old and New Testaments. Also, Jewish couples of modest income offered two turtle doves instead of the customary lamb as a sacrifice to God when they presented their newborn child in the Temple.
Learn more about turtle doves at Garden-birds.co.uk.