Purgatory is a state of being after death in which the soul is cleansed of any stain from sins committed after baptism by the working of the God.
It is often described as akin to a fire purifying gold.
It is temporary; it is outside of time.
The Church at one point said indulgences would free people of so much "time" from purgatory, but has generally gone away from that due to the inability for us to really understand a concept of time after death (and there's also a theological issue of those who haven't done their "time" after the eschaton).
So, purgatory is a state of being in which the soul is in the process of becoming fully purified so as to be in the light of God without blemish. We cannot really say for certain (in the positive) more than that.