I agree with and appreciate all the replies.
The descendants of Abraham, Isaac and Jacob were the Jews who lived in Israel and were considered "God's Children" and who became God's children by the Faith of Abraham towards God and also because of God's promise to Abraham. To keep the very same promise to Abraham's descendants God sent His only begotten Son Lord Jesus Christ to earth. So, initially Lord Jesus Christ was the "Children's Bread" sent from heaven to the Jews, to minister to the Jews and save them.
The remaining people who were called Gentiles are indirectly being referred to as Dogs. This woman whose daughter was grievously vexed with a devil was a Gentile too. Lord Jesus Christ wanted to test her faith and humbleness by such harsh words. Initially, He does not even reply to her. His Disciples ask Him to send her away for she was still coming behind them. But he answered and said, "I am not sent but unto the lost sheep of the house of Israel". Then she again comes to Him and worships Him and cries "Lord, help me". Still the Lord tests her by saying, "It is not meet to take the children's bread, and to cast it to dogs". This could have been a humiliating experience for any person. This strange behaviour of Lord Jesus Christ could have been thought as a lack of politeness or arrogant behaviour by people. But, this woman did not have any such thoughts about our Lord. And, she did not loose her hope and was stubborn/persisting enough to get her blessing from the Lord. Also, she humbled herself by acknowledging that she could indeed be referred to as a Dog and said "yet even dogs eat the crumbs from their master's table". All this could have melted the heart of Lord Jesus Christ. Then Jesus answered and said unto her, O woman, great is thy faith: be it unto thee even as thou wilt. And her daughter was made whole from that very hour. He compliments her lavishly by calling her as a woman of great faith and fulfills the desire of the woman by making her daughter perfectly allright again.
One can learn a few things from this true story especially from the woman. She did not loose her hope. She was persistent enough. She was extremely humble enough. She had tremendous love for her daughter. Her love for her daughter made her to do all this. Whenever i read this story, i feel i am worse than this woman.
Prior to this incident Lord Jesus Christ sends His twelve disciples to the lost sheep of Israel (meaning Jews or descendants of Abraham). And He also says not to go to the Gentiles or the Samaritans. This clearly shows that the initial plan of Lord Jesus Christ was to preach the gospel of salvation and conduct all kinds of miracles to the Jews only. The verse below gives the clue -
Matthew 10:5 - These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not:6 But go rather to the lost sheep of the house of Israel.
Apostle Paul explains through the verses below that since most of the Jews rejected their Messiah Lord Jesus Christ and did not believe that He was sent from Father-God, Salvation came to the Gentiles also, so as to provoke the Jews to feel jealous and make them realize their mistake. So, blindness in part has happened to Israel until the fulness of Gentiles comes in. It is also prophesied by Prophet Isaiah that Lord Jesus Christ shall come from the generations of Jesse (Father of King David) and in Him shall the Gentiles trust.
Romans 11:11 - I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy.
Romans 11:25 - For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in.
Romans 15:12 - And again, Esaias saith, There shall be a root of Jesse, and he that shall rise to reign over the Gentiles; in him shall the Gentiles trust.