"But instead, one brother goes to law against another - and this in front of unbelievers!" NIV translation
Not only does this affect the harmony of believers but also unbelievers are watching us at all times. When they see us acting in the same ways that they do, it does not glorify God and make them want to know about Jesus.
"The very fact that you have lawsuits among you means you have been completely defeated already. Why not rather be wronged? Why not rather be cheated? Instead, you yourselves cheat and do wrong, and you do this to your brothers." NIV translation
Paul tells them (and us) that the fact that we even have lawsuits among believers is a sign that we are not following Christ. Christ came as a servant and was wronged many times including when "He who knew no sin became sin" for us. Even while He was being beaten, mocked, and crucified, He asked His Father not to hold their actions against them. That, my friends, is the standard for us as well and anything less is defeat.
"Do you not know that the wicked will not inherit the kingdom of God? Do not be deceived: Neither the sexually immoral nor idolators nor adulterers nor male prostitutes nor homosexual offenders nor thieves nor the greedy nor drunkards nor slanderers nor swindlers will inherit the kingdom of God. And that is what some of you were. But you were washed, you were sanctified, you were justified in the name of the Lord Jesus Christ and by the Spirit of our God." NIV translation
Here, Paul gives a list of sins that were common in the city of Corinth (the "sin city" of the Roman Empire). In giving us this list, he is helping us to see that the unforgiveness that leads to these lawsuits are sinful just as all of the things on the list. He goes on to remind us that we were forgiven for the sin that we were in and so we are called to forgive those who have wronged us.
"'Everything is permissible for me' - but not everything is beneficial. 'Everything is permissible for me' - but I will not be mastered by anything." NIV translation
For all of those who walk around afraid that they are going to slip up, sin, and lose their salvation, this verse should bring comfort. Paul tells the people in this "sin city" that they have the freedom to do anything (and not lose their salvation) but not everything will help them in their walk with the Lord. He describes how a taste of sin can lead to control by that sin instead of control by the Spirit of God.