1 Cor. chapter 15 deals with the dead. In the Catholic Tradition, we believe that when a person dies, he/she is either sent to Heaven or hell. When one goes to Heaven, that person must first go through a state called Purgatory. People who are in this state are suffering, because parts of them are being removed, and others perfected. We pray for the dead that God may have mercy on them and that their time in purgatory is shortened, or at least as painless as possible. So Chapter 15 pretty much explains how we pray and why. 1 Cor. 3:10-15 details what exactly goes on in Purgatory. The fire is the means by which we are cleansed and perfected. As the bible says, nothing impure can enter Heaven. Therefore, we must be purified and made perfect, and this transformation is called Purgatory.
[1 Cor. 3:10-15](http://www.usccb.org/nab/bible/1corinthians/1corinthians3.htm) and [1 Cor. 15](http://www.usccb.org/nab/bible/1corinthians/1corinthians15.htm) Both are from the NAB (New American Bible)