When the Gentile woman says, "Lord, help me," Jesus up to this point had spoken only to His disciples. Now He speaks to the woman, telling her she is not of Israel and that, "It is not good to take the children's bread and throw it to the little dogs." By "children" He means Israelites (Acts 10:36), while "dogs" were symbols of unclean Gentiles, a proverbial expression used by the Jews to represent their sense of national superiority over the nations.
Jesus does not Himself call the Gentiles "dogs," using the term only here to point out the normal antipathy between Jews and Gentiles, which His disciples had echoed. The word He uses for "dogs" is a mild one, meaning "little dogs" or "puppies"—not large, wild dogs native to the area but domesticated animals like those the Romans had introduced during their occupation. It suggests the family puppy under the table at dinnertime, begging for a scrap.
Because of her faith and humility, the woman does not take offense at this. His words do not discourage her because she was hopeful with faith, and her works demonstrate that hers was not a dead faith, but a strong one. She was resourceful and knew enough about Jesus to believe that He was both compassionate and powerful. Feeling deeply unworthy and contentedly accepting her place among the dogs, she merely asks for spiritual crumbs from His merciful table—a little crumb for her daughter is all she seeks.
Counting herself a "puppy," she faithfully looks forward to being counted by God as His child (Galatians 3:26). Although she stands outside of the elect family of Israel, she trusts that Jesus' goodness would impart a blessing. By intervening on behalf of her and her daughter, Jesus shows that the Gentiles' potential for salvation is no less than that of Israelites.