Matthew was writing to a Hebrew audience and one of the purposes of his Gospel was to show from Jesus' genealogy and fulfillment of Old Testament Prophecies that Jesus was the long-expected and promised Messiah, and thus should be believed on. Matthew's emphasis is upon Jesus as the Messiah or promised King, the "Son of David" who would forever sit upon the throne of Israel.
Mark, a cousin of Barnabas (Colossians 4:10), was an eyewitness to the events in the life of Christ as well as being a friend of the Apostle Peter. Mark wrote for a Gentile audience as is brought out by his not including things important to Jewish readers (genealogies, Christ's controversies with Jewish leaders of His day, frequent references to the Old Testament, etc.). Mark emphasizes Christ as the suffering Servant, the One who came not to be served but to serve and give His life a ransom for many (Mark 10:45).
Luke, the "beloved physician" (Colossians 4:14), evangelist, and companion of the Apostle Paul, wrote both the Gospel of Luke and the Acts of the Apostles. Luke is the only Gentile author of any of the New Testament. He has long been accepted as diligent and master historian by those who have used his writings in geological and historical studies. As a historian, he states that it is his intent to write down an orderly account of the life of Christ based on the reports of those who were eyewitnesses (Luke 1:1-4). Because he specifically wrote for the benefit of Theophilus, apparently a Gentile of some stature, his gospel was composed with a Gentile audience in mind, and his intent is to show that a Christian's faith is based upon historically reliable and verifiable events. Luke often refers to Christ as the "Son of Man," emphasizing His humanity and shares much detail that is not contained in the other Gospel accounts.
The Gospel of John, written by John the Apostle, is distinct from the other three gospels and contains much theological content in regards to the person of Christ and the meaning of faith. Matthew, Mark, and Luke are often referred to as the "Synoptic Gospels" because of their similar styles and content. The Gospel of John begins not with Jesus' birth or earthly ministry but with the activity and characteristics of the Son of God before His becoming man (John 1:14). The Gospel of John emphasizes the Deity of Christ as is seen in his use of such phrases as "the Word was God" (John 1:1), "the Savior of the World" (4:42), the "Son of God" (used repeatedly), "Lord and...God" (John 20:28 ) in describing Jesus. In John's Gospel, Jesus also affirms His Deity with several "I Am" statements, most notable among them is John 8:58, in which He states that "...before Abraham was, I Am" (compare to Exodus 3:13-14). But John also emphasizes the fact of Jesus' humanity, desiring to show the error of a religious sect of his day, the Gnostics, who did not believe in the humanity of Christ. John's spells out his overall purpose for writing towards the end of his gospel: "And truly Jesus did many other signs in the presence of His disciples, which are not written in this book; but these are written that you may believe that Jesus is the Christ, the Son of God, and that believing you may have life in His name" (John 20:30-31)(NKJV).