Both Orthodox and Catholic believe that upon death few souls are ready to enjoy the vision of God, so called beatific vision and the highest state of happiness, to which we refer as Heaven or Paradise. We both believe that there is an intermediate state after death and then two possible final states, Heaven or Hell. We both believe that prayers should be offered for the repose of the souls of the dead, and that for those in an intermediate state these prayers have a salutary effect.
The Catholics believe that Christ judges everyone in so-called particular judgement immediately after death and if the soul is condemned to Hell, that is where it goes right away. If the soul is judged saved for eternal life, its final destination is heaven. However, most souls are burdened by the aftereffects of sin, even though that sin is forgiven them. The analogy is that even though a tumor has been removed from a sick man (=sin pardoned), the scar where the surgery took place needs to heal before the man can walk again (=be in the presence of God). That convalescence or purification happens in Purgatory. One mark of a saint is that he or she has completed the purification on earth in his lifetime and can go directly to heaven. The prayer is only efficacious for souls in purgatory, but not knowing who ends up where and at what time, we pray for all dead (a public mass for the dead is not allowed in cases of hardened public sinners, but private prayers are always a good thing).
The Orthodox beliefs are somewhat different. Both the condemned and the saved go through a process of trial, when Satan and Christ bargain for the soul. This is when we pray for them. The Orthodox do not use the word purgatory and consider the Catholic doctrine an innovation. I don't think they would say that our belief is definitely wrong, but it is not officially taught by the Orthodox Church.
You may be interested in some in-depth discussion I recently had with the Orthodox so that we understand each other better: Indulgences: Spreading the Wealth. I did not intend it as a vehicle for polemics witht he Protestants and you will see that I avoid arguing the scriptural basics there.
It is true that the doctrine of Purgatory, even its Orthodox less detailed counterpart is in some part speculative theology, in some part tradition that was written down by the Church fathers but never made it into the New Testament proper, and in some part scripture. I showed you why the Catholic theology of the afterlife is compatible with the scripture, but I agree that different interpretation of the same scripture is possible. Certianly the Orthodox interpretation is possible even though it differs from ours. I don't think that any interpretation of the scripture that denies the existence of an intermediate stage following death is possible though; at least I have not seen a satisfying exegesis of the parable of Unmerciful Debtor that would deny that a temporal punishment after death exists for some people.
Here's a quick synopsis of the belief of the Orthodox Church on the state of souls after death from the GOA website. I believe it was written by Archbishop Michael:
"At death man's body goes to the earth from which it was taken, and the soul, being immortal, goes to God, who gave it. The souls of men, being conscious and exercising all their faculties immediately after death, are judged by God. This judgment following man's death we call the Particular Judgment. The final reward of men, however, we believe will take place at the time of the General Judgment. During the time between the Particular and the General Judgment, which is called the Intermediate State, the souls of men have foretaste of their blessing or punishment."