A Greek - The Jews called all persons "Greeks" who were not of their nation. Compare Romans 1:14. The whole world was considered as divided into Jews and Greeks. Though she might not have been strictly a "Greek," yet she came under this general appellation as a foreigner.

The woman was a Greek - Rosenmuller has well observed, that all heathens or idolaters were called Ἑλληνες, Greeks, by the Jews; whether they were Parthians, Medes, Arabs, Indians, or Ethiopians. Jews and Greeks divided the whole world at this period.

The woman was a (m) Greek, a (n) Syrophenician by nation; and she besought him that he would cast forth the devil out of her daughter.
(m) By nationality, profane.
(n) A neighbour of or near to Damascus.

The woman was a Greek,.... Or Gentile, an Heathen woman, which made her faith the more remarkable. So the Syriac, Persic, and Ethiopic versions call her; which she might be, and was, though she was a woman of Canaan, as she is said to be in Matthew 15:22, for though the land of Israel in general, was called the land of Canaan, yet there was a particular part, which was at first inhabited by Canaan himself, which bore this name; and is the same with Phoenicia, of which this woman was an inhabitant, and therefore she is afterwards called a Syrophoenician; See Gill on Matthew 15:22. And this place was now inhabited by Gentiles; hence the Jews often distinguish between an Hebrew and a Canaanitish servant; of which take an (z) instance or two;
"an Hebrew servant is obtained by money, and by writing, a Canaanitish servant is obtained by money, and by writing, and by possession.''
Again (a),
"he that does injury to an Hebrew servant, is bound to all these (i.e. to make compensation for loss, pain, healing, cessation from business, and reproach), excepting cessation from business--but he that hurts a Canaanitish servant, that belongs to others, is bound to them all.''
And by a Canaanitish servant, they understand any one that is not an Israelite; for an Hebrew and a Canaanite, are manifestly opposed to one another. This woman being of Phoenicia, as appears by what follows, which was sometimes called Canaan, might be said to be a woman of Canaan, and also a Gentile.
A Syrophoenician by nation; or extract. The Syriac and Persic versions say she was "of Phoenicia of Syria"; and the latter, by way of explanation, "of Emisa". The Arabic version adds, "her extraction was of Ghaur"; and the Ethiopic version says, she was "the wife of a Syrophoenician man"; See Gill on Matthew 15:22.
And she besought him, that he would cast forth the devil out of her daughter; which she was persuaded, by what she had heard of him, he was able to do, by a word speaking, though her daughter was not present.
(z) Misn. Kiddushin, c. 1. scct. 2, 3. (a) Misn. Bava Kama, c. 8. sect. 3.

The woman was a Greek--that is, "a Gentile," as in the Margin.
a Syrophœnician by nation--so called as inhabiting the Phœnician tract of Syria. JUVENAL uses the same term, as was remarked by JUSTIN MARTYR and TERTULLIAN. Matthew (Matthew 15:22) calls her "a woman of Canaan"--a more intelligible description to his Jewish readers (compare Judges 1:30, Judges 1:32-33).
and she besought him that he would east forth the devil out of her daughter--"She cried unto Him, saying, Have mercy on me, O Lord, Son of David: my daughter is grievously vexed with a devil" (Matthew 15:22). Thus, though no Israelite herself, she salutes Him as Israel's promised Messiah. Here we must go to Matthew 15:23-25 for some important links in the dialogue omitted by our Evangelist.
Matthew 15:23 :
But he answered her not a word--The design of this was first, perhaps, to show that He was not sent to such as she. He had said expressly to the Twelve, "Go not into the way of the Gentiles" (Matthew 10:5); and being now among them Himself, He would, for consistency's sake, let it be seen that He had not gone thither for missionary purposes. Therefore He not only kept silence, but had actually left the house, and--as will presently appear--was proceeding on His way back, when this woman accosted Him. But another reason for keeping silence plainly was to try and whet her faith, patience, and perseverance. And it had the desired effect: "She cried after them," which shows that He was already on His way from the place.
And His disciples came and besought Him, saying, Send her away; for she crieth after us--They thought her troublesome with her importunate cries, just as they did the people who brought young children to be blessed of Him, and they ask their Lord to "send her away," that is, to grant her request and be rid of her; for we gather from His reply that they meant to solicit favor for her, though not for her sake so much as their own.
Matthew 15:24 :
But He answered and said, I am not sent but unto the lost sheep of the house of Israel--a speech evidently intended for the disciples themselves, to satisfy them that, though the grace He was about to show to this Gentile believer was beyond His strict commission, He had not gone spontaneously to dispense it. Yet did even this speech open a gleam of hope, could she have discerned it. For thus might she have spoken: "I am not SENT, did He say? Truth, Lord, Thou comest not hither in quest of us, but I come in quest of Thee; and must I go empty away? So did not the woman of Samaria, whom when Thou foundest her on Thy way to Galilee, Thou sentest away to make many rich! But this our poor Syrophœnician could not attain to. What, then, can she answer to such a speech? Nothing. She has reached her lowest depth, her darkest moment: she will just utter her last cry:
Matthew 15:25 :
Then came she and worshipped Him, saying, Lord, help me!--This appeal, so artless, wrung from the depths of a believing heart, and reminding us of the publican's "God be merciful to me a sinner," moved the Redeemer at last to break silence--but in what style? Here we return to our own Evangelist.

The woman was a Greek (that is, a Gentile, not a Jew) a Syrophenician or Canaanite. Canaan was also called Syrophenicia, as lying between Syria, properly so called, and Phenicia.