Praying for Souls in Purgatory

November 2nd is All Souls Day and the Catholic Church encourages the faithful to pray for souls in Purgatory. What do Catholics actually believe Purgatory is and why are Catholic faithful encouraged to pray for souls going through purification?

References
Gospel according to Mathew 5:22-26
Gospel according to Mathew 12:32
Book of Revelations 21:27
1st letter of Saint Paul to the Corinthians 3:11-15
Book of Job 1:5
1st letter of Saint Paul to Timothy 2:1-4
Letter from James 5:16
2nd Book of Maccabees 12:39-42, 46
Catechism of the Catholic Church Paragraphs 1030-1032