a little dog: a small dog kept in the house as a family pet, not a stray or scavenging street dog
This word is a smaller, homier form of the ordinary Greek word for a dog (κύων). It appears only four times in the New Testament, and all four are in a single scene: a Gentile woman begs Jesus to heal her daughter, and he answers that it is not right to take the children's bread and throw it to the little dogs (Matthew 15:26; Mark 7:27). She replies that even the little dogs under the table eat the crumbs that fall from the children (Matthew 15:27; Mark 7:28).
In every surviving example of the word in ancient Greek, it names an indoor pet, the small dog that lived in the house and waited under the dining couch, never a wild dog of the streets. So the picture Jesus draws is a household one: a family, its children, and the pet dogs at their feet. This matters, because the harsher image some readers assume, a filthy scavenger snarling in the alley, is not what the word says.
A house pet, however, is still not a child. Most scholars now agree that the gentler, domestic picture does not turn the saying into a compliment: the woman is being placed on the dog's side of the household rather than the children's, and children are fed first. The force of the exchange is about priority, Israel before the nations, not about calling the woman a cherished pet. Her famous answer accepts the picture and presses it: pets eat at the same time as the children, right under the same table.