One could spend much time tracing the genealogical records of Christ from generation to generation, learning as much as possible about each one mentioned therein. Such will not be our purpose at this time. Rather, we will endeavor to make a few observations that we hope will be of interest and value from Matthew 1:1-17 and also Luke 3:23-38.
First, it should be noted that Matthew, as he traces the record from Abraham to Christ, mentions only four women: Tamar, Rahab, Ruth, and Bathsheba (though she is only mentioned indirectly). Is there a specific purpose in naming only these women and neglecting many others who could have easily been mentioned (e.g., Sarah, Rebekah, Rachel, etc.)? I believe the answer is "yes." Matthew's purpose is not to defame these women, but rather to place emphasis upon them in order to demonstrate the fact that all people, regardless of background, are of concern to God. Tamar committed incest; Rahab was a prostitute; Ruth was a despised Moabitess; and Bathsheba committed adultery. Yet the Redeemer of the world, the true King of Israel, had these women (as well as many men) in His ancestry who were not always what they should have been. God can (and does) use sinful people to accomplish His will. The inclusion of these women on this list also serves as a reminder that Jesus was the only sinless person who ever lived, and He came for the benefit of all mankind.
Second, it should be observed that Matthew traces Jesus' genealogy through Joseph, who was the legal father of Jesus. Although such seems strange to us, since Jesus was not begotten from Joseph, the Jews emphasized the father's lineage. Luke, on the other hand, traces Jesus' lineage backwards. He states that it "was supposed" that Jesus was the son of Joseph (Luke 3:23)--though Luke had already proved this false in a technical sense (1:26-35). Joseph is said to be "the son of Heli" (3:23). This does not contradict Matthew 1:16 where he is said to be the son of Jacob. Luke is writing to the Greeks who would care nothing about Joseph's ancestry since he wasn't the literal father of Jesus. It appears that Joseph was counted as "the son of Heli" in the sense that he married Heli's daughter, Mary. Therefore, Luke is showing Jesus' ancestry through Mary.
Thus, Matthew's genealogy, through Joseph, does not prove Jesus to be of the blood of David, because Jesus was not related to Joseph through blood. However, Joseph was Jesus' legal father, and thus, Matthew's record does show Jesus as a possible heir to the throne of David. Luke's genealogy, through Mary, does not prove Jesus to be an heir of David's throne, but it does show Jesus to be of the blood of David.
In summary, these genealogies in Matthew and Luke: (1) provide absolute certification of Jesus' ancestry, (2) emphasize the humanity of Jesus, (3) proclaim the faithfulness of God in fulfilling His promise to the patriarchs, (4) hint at the sinfulness of man, (5) show God's purposes in His dealings with Israel, (6) support and complement the virgin birth narrative, and (7) complement each other in an interesting way. God's word is so rich in every passage--even the genealogies!