The Bible says nothing at all about purgatory.
The Catholics and some Eastern churches say it is a place where people go when they die, to be purified from their venial excusable, pardonable sins.
Catholic doctrine says that this cleansing in purgatory is by fire. It seems to have come about through a belief that all would gain salvation but some had to be punished first. One Bishop Gibson says that belief in purgatory was not known in the Church for a thousand years ("Preservative against Popery", vol viii. p 255) (This info is from the Encyclopedia Britannica). There are many claims that the Roman Catholic church frequently used the cases of people supposedly in purgatory to make their families pay money to get them out of that situation and into heaven.
The belief in purgatory goes against all that the Bible tells us. The Bible says that Jesus Christ was crucified so that we could have our sins forgiven through him (1 Peter 2:24)
"He himself bore our sins in his body on the tree, so that we might die to sins and live for righteousness; by his wounds you have been healed".
We are told that our sins will be forgiven if we repent and ask for forgiveness. (see also 1 John 2:12)
The Bible does not allow for a period of time of trial by fire, but says rather that when we die we lose all knowledge of everything.
Ecclesiastes 9:5 tells us, "For the living know that they will die, but the dead know nothing." see also verse 10, and Psalm 146: 3,4.
The Bible states that death is later followed by resurrection when Jesus Christ returns to the earth.
John 5:28,29 reads, "Do not be amazed at this, for a time is coming when all who are in their graves will hear his voice and come out - those who have done good will rise to live, and those who have done evil will rise to be condemned."
Hades and Paradise are associated beliefs. The word 'hades' actually means 'unseen world' or, in the Bible context, the grave. 'Paradise' means 'a grand reserve, or park' (think of the Garden of Eden) though it later came to be associated with purgatory.
Jesus said through John in Revelation 2:7:
"To him who overcomes, I will give the right to eat from the tree of life which is in the paradise of God."