Souls in purgatory experience two forms of punishment: the pain of loss and the pain of sense.
The pain of loss is a delay in the vision of God. As the souls in purgatory are interested in God and not in created goods anymore, their desire to possess God is extremely intense. But they cannot enjoy Him until they have totally expiated their faults and the punishment of their sins.
The pain of sense, on the other hand, is referred to in several documents of the Magisterium as fire. This fire’s way of operating is mysterious though. Also, St. Augustine wrote that purgatory’s pain of sense is more severe than anything man can suffer here on earth.
The purpose of the punishment in purgatory is to cleanse the soul so that it can reach the total purity required to enjoy the Beatific Vision.
It is hard to imagine that there is joy in purgatory. But it’s true – purgatory is not a sad “place.” It is rather the opposite, in spite of the pains. Although the souls suffer greatly, their joys and consolations are beyond description. This is because they know that they are sure to enter heaven; they are comforted and they enjoyed the spiritual help of the Blessed Virgin, the angels and the saints. Moreover, their pains can be reduced through our help, most importantly through the Holy Mass. We can also offer sacrifices and penances on their behalf, gain indulgences for them, pray for them, and give alms in their memory.
(Reference: “The Last Things” by Carlos Escartin in “Faith Seeking Understanding Vol. II”)