The Catholic Church believes that each person has a soul which gives them life, and this soul will live forever. After a person's death, their soul will continue to exist, and then at the resurrection of the dead at the end of the world, their soul will be re-united with their resurrected body and will live forever.
The Immortal Soul in the Bible
The Bible speaks in several places about the soul as being distinct from the body and living forever. “And do no fear those who kill the body but cannot kill the soul; rather fear him who can destroy both soul and body in hell.” - Matthew 10:28. In this passage, Jesus speaks of the distinction between soul and body, and that it is more important to take care of one's soul than to take care of one's body, although both are important. St. Paul also recognizes this distinction between body and soul. “May the God of peace himself sanctify you wholly; and may your spirit and soul and body be kept sound and blameless at the coming of our Lord Jesus Christ.” - 1 Thessalonians 5:23. Furthermore, the soul lives on after death, apart from the body, and lives forever. “But the souls of the righteous are in the hand of God, and no torment will ever touch them. In the eyes of the foolish they seemed to have died, and their departure was thought to be a disaster, and their going from us to be their destruction; but they are in peace. For though in the sight of others they were punished, their hope is full of immortality.” - Wisdom 1 – 4. The author of this book speaks of souls and their hope of living forever, even after death. It defies those who see death as the end of everything, for the soul has a life after this world.
St. Polycarp on The Immortal Soul
The early church fathers did not speak much on this subject, but St. Polycarp had a beautiful reflection on the immortality of the soul.
I bless you for because you have considered me worthy of this day and hour, that I might receive a place among the number of martyrs in the cup of your Christ, to the resurrection to eternal life, both of soul and of body, in the incorruptibility of the Holy Spirit - Martyrdom of Polycarp 14:2
As he was about to die, Polycarp professes his belief that the soul is immortal and that because of his martyrdom, he will be joining God in Heaven to live there forever. If he did not hold this belief in the immortality of the soul, accepting martyrdom would be foolish, for there would be no reward in an afterlife and it would be cutting short this current life on earth for no reason. Belief in the immortality of the soul is what drives the saints to cheerfully accept their death in whatever form it comes.