Basic truths about the Pope
The papacy was instituted by Christ. He chose St. Peter as the head of the Apostles and of His new Church. Out of his closest followers, He needed one to be the leader and His visible delegate on earth. Peter eventually became the bishop of Rome, and his successors continued his role as the supreme head of the Catholic Church. This is why the pope is also known as the bishop of Rome. The line of popes can be traced back from our pope now to St. Peter. This consistent and unbroken succession is a sign of the apostolic nature of the church, by which our pope and bishops can be shown to be direct successors of the Apostles who received their commission directly from Jesus.
On a very practical level, the pope is a necessary and useful institution because he is a sign and keeper of unity. Without a clear leader, the church would not keep to one path and remain united. There will always be differences of opinion, but without one person with whom all disputes must ultimately be resolved, everyone will split. If there is this one person appointed leader, then the church will remain united under that one.
Pope Francis
Our current pope is Pope Francis. He was elected in March 2013. He was the bishop of Buenos Aires, Argentina before his election. He is a Jesuit, the first one to become pope. While bishop, and now as pope, he has had a special love for the poor and wanted to make sure that the least of society are not forgotten or neglected. He has also had a special concern for those who feel abandoned by the Church. He wants to show God's mercy in tangible ways so as to encourage all to return to grace and Christ.
As his papacy is still relatively new, he does not have a long list of writings, but he does have some significant works to mention. He has written the Apostolic Exhortation Evangelii Gaudium about proclaiming the gospel in all parts of the world. He has also written two encyclicals. The first Lumen Fidei, was written in conjunction with Pope Benedict XVI and spoke of the need for faith. His second Laudato Si spoke of the need for caring for the earth and its gifts as God intended for us to do. These are important to read in order to gain a greater understanding of what he is trying to teach us.
Papal Infallibility
One common source of confusion about the pope is the Catholic claim of infallibility. Catholics believe that the pope if infallible, namely that he can only truth in matters of faith and morals. Some people confuse this with impeccability, meaning free from sin. However, the pope is just as human as every other person and just as liable to sin as any other person. There have been plenty of very terrible and sinful popes, and yet not one of them has ever taught something at odds with the doctrine of the Church. This is really quite amazing. It also doesn't mean that whatever he says won't be wrong. Infallibility only applies to matters of faith and morals. If the pope makes a definitive statement, declaring as pope that some action is morally wrong, then that is so. The Holy Spirit guarantees that that statement is true. However, if the pope makes some statement about any other subject, like science, then his statement is as subject to error as that of any other person. The concept of infallibility is very limited, but within its limits, it is very powerful and reassuring, for people can know that his pronouncements are indeed what God Himself would tell them, namely the truth.
Was Peter the first Pope?
Some people argue that there is no evidence in the Bible that Peter was the head of the Apostles and the one chosen by Jesus to be the leader. They point to Peter being rebuked by Paul in Galatians. Paul was rebuking Peter on a matter of practice and accusing him of hypocrisy. This also has to do with infallibility and impeccability. Just because Peter was pope does not mean that he could not make mistakes, and Paul was pointing out a mistake that he perceived in Peter's life, namely that he was causing scandal by not eating with the Gentiles. This did not concern a matter of doctrine, but rather of personal behavior, so Peter could have been wrong and still been pope and infallible.
There are also many arguments for and against the passage in Matthew 16 as to whether it shows that Peter was intended to be the leader of the Apostles or not. Many of these arguments depend upon technical interpretations of Greek words, so unless one knows Greek, a person is just taking the word of dueling scholars. There is one argument, though, that does show Christ meant Peter to have a special function. In the Bible, whenever someone's name is changed, it is to show a special mission from God, as in Abram to Abraham or Jacob to Israel. Peter's original name was Simon, and Christ re-named him Peter in the above passage in Matthew. He does not change the name of any other person in the New Testament, and so this new name which he gives to Simon/Peter does show that this particular apostle was to have a special mission and a different level of responsibility or authority than the others.
Some people also argue that it is impossible to know if the Bishop of Rome is supposed to be the Pope, since there is no clear evidence in the Bible that Peter was ever in Rome and that he specifically wanted to hand down his ministry to the bishops of that city. This view has a lot to do with the Catholic understanding of the Bible and Tradition as the dual-approach to doctrine, as opposed to relying on the Bible alone. If, as the Church teaches, Sacred Tradition is as important as the Bible in looking at doctrines and determining truth, then we don't have to look exclusively at the Bible to determine whether the Bishop of Rome is indeed supposed to be the supreme head of the Church. There is evidence from the writings of the Church Fathers that the bishop of Rome was considered to be the head and final arbiter of decisions. There are many of them, from the first four centuries or so after Christ's death, who wrote of both Peter and the Bishop of Rome as the head of the Christian Church.