In Catholicism, a big part of baptism is the idea that it washes away original sin. As I understand things Eastern Orthodoxy denies original sin, so I'm wondering what the significance of baptism is for an Eastern Orthodox. I've read in passing that there is some idea called "ancestral sin" which is analogous to the Catholic concept of "original sin": Does baptism wash this sin away?
My understanding is that ancestral sin is different from original sin in that ancestral sin teaches we have received a physiology, psychology, and environment that is deeply damaged by sin and therefore bound by it at birth, through no fault of our own, and have a great inclination towards it; whereas the West and Reformed theology specifically likes to emphasize how guilty we are and how justly we deserve punishment, even for the sins we didn't commit. So the basic problem that the church is trying to solve is different in each place. In Orthodoxy the human is sick and in Catholic and Protestant teaching the human is guilty. In both teachings baptism gives us a remission of sins, but it is understood differently, because it is solving different problems.
This is probably the general explanation that you will hear from most any Orthodox apologist (the sick vs. guilty comparison), but that is not to say that each theology doesn't overlap. There are certainly teachings of guilt and healing in each tradition. However, I think you question betrays a common Western question about Orthodoxy: namely that if salvation isn't understood within a legal construct of guilt and payment then how can any of it make sense.
In other words, yes, baptism (and the mysteries/sacraments in general) wash away sin, but it is not the legal condemnation of God that is being done away with. We didn't need to fulfill, or have fulfilled for us, some part of God's law to satisfy any sort of justice. It is our own soul and body that is being transformed by the Holy Spirit to enable communion.