Essentially, Paul's first letter to the Corinthians largely addressed problems and divisions in the church. A few years after establishing the church and then leaving, Paul hears that those left in charge had basically developed some spiritual "arrogance" and as a result, were confusing the basic teachings of Christ.
What is central to Paul's message then is this: Christ came as the one who died, was buried, and raised to new life as the atonement for human sins. As Adam was the first man to fall, Christ came as the "last Adam," meaning his death and resurrection allowed for humanity to be separated from God no more. As a result, Jesus' message was one of love and unity. Jesus came to bring equal forgiveness for all believers and would have his church unified.
Is Christ divided? Was Paul crucified for you? Were you baptized into the name of Paul? (1 Cor 1: 13)
Additionally, speaking to the arrogance displayed in the church leadership, Paul adds that the church should be relying on the teaching and wisdom of Christ for answers, rather than on each other. Christ is the representation of God's wisdom:
but to those whom God has called, both Jews and Greeks, Chrst the power of God and the wisdom of God. (1 Cor. 1: 24)
My message and my preaching were not with wise and persuasive words, but with a demonstration of the Spirit's power, so that your faith might now rest on men's wisdom, but on God's power. (1 Cor 2: 4-5)
With this key principle established (that it is by God's authority through Jesus and not man), Paul goes on in the rest of the letter to address specific problems the Corinthian church was having.