Basically, we die. When we die, our souls get a foretaste of the life to come which is basically either rest, or torment - sometimes when someone's passed away we'll refer to them as having "fallen asleep" i.e. "Fr John has fallen asleep in the Lord". This is temporary and, from what I remember, more or less based on the condition of one's soul when you died. We're in this temporary state until the Second Coming when the Lord resurrects the dead which means that your soul is now reunited with your body (part of why we're against cremation). At some point we'll have the Last Judgement. Everything else beyond that basically comes down to mystery, or theological opinion, but there are some things that we know just aren't so and the West's idea of purgatory just isn't one of them.
We pray for the dead partly to help them out in the afterlife and partly for our own comfort because generally we miss relatives and friends after we die. Part of the story of St Xenia of St Petersburg says that her husband, who was an Imperial Russian Army officer, had died in a not-so-good spiritual state. Being a good widow and good saint she prayed for his soul for decades. After decades of praying for his soul, the Lord revealed to her that his soul was saved due to her prayers.