Although firmly settled in the land that we know today as Israel, Abraham wished for his son Isaac to marry among his own people, so he sent his chief servant back to Ur (see The Tigris-Euphrates Valley) to find him a wife (Genesis 24:1-9).
Once there, the servant, no doubt with God's help, immediately found Rebekah, who agreed to return with the servant (Genesis 24:10-65). Rebekah didn't simply leave to go live with complete strangers. She was the granddaughter of Abraham's brother (Genesis 24:48), and therefore Isaac's second-cousin. They were married soon after her arrival in Israel (Genesis 24:67). Isaac was then 40 years old (Genesis 25:20), Rebekah was much younger, perhaps late-teens or early twenties (Genesis 24:16)
Eventually the marriage produced two children - twin boys, Jacob and Esau. They were fraternal, not identical, twins - they were very different in appearance (Genesis 27:11), and personality (Genesis 25:27). As with Isaac and his brother Ishmael, the conflict and competition between Jacob and Esau lasted their entire lives. There was no love between them. Part of it may have been due to blatant favoritism on the part of the parents as they were growing up - "Isaac ... loved Esau, but Rebekah loved Jacob." (Genesis 25:28).
When the time came for the then-elderly Isaac to give his blessing (in effect, a passing of family leadership) to his oldest son, which happened to be Esau (by a few minutes), Rebekah devised a plan to have Jacob get the blessing instead. Jacob had already bought Esau's birthright for that now-famous bowl of stew (Genesis 25:27-34).
While Esau was out hunting for the wild game that Isaac had asked for before giving his blessing, Rebekah had Jacob disguise himself as Esau, to get the blessing from his nearly-blind father. The scheme worked - Jacob then had both the birthright, and the blessing (Genesis 27:1-29).
When Esau returned from the hunting trip, what Rebekah and Jacob had done was discovered. Isaac and Esau were outraged, but it was too late. Even though it had been obtained by an unintended recipient through deception, the blessing could be not given twice (Genesis 27:30-40).
Isaac apparently was willing to let the matter stand, but Esau had something else in mind. He planned to murder his brother Jacob, perhaps out of sheer rage, or maybe as a way to recover the blessing and birthright which would then pass back to him. Rebekah however heard about what Esau was planning, and sent Jacob back to live with her brother Laban (Genesis 27:41-45). It was while on that journey that Jacob had his Stairway To Heaven dream - what Rebekah and Jacob had done was actually God's will.
Rebekah had hoped that Jacob's living far away would only last until Esau's fury subsided (Genesis 27:44), but it was the last she ever saw of him - she died before his return many years later.
Rebekah was buried in Hebron in a family tomb where Abraham, Sarah, Isaac, Jacob and Leah were all buried. Today, the place is known as the Tomb of The Patriarchs.
Fact Finder: Is Rebekah mentioned in the New Testament?
Note: Romans chapter 9 explains God's sovereign choice.