Without a doubt the most important religion in the Middle Ages in the West was Christianity, though perhaps not in the world. As for features, there is some room for debate, but I would have to say that one of the most important and lasting features of the church in the Middle Ages is the idea of authority. More specifically, it is the idea that all authority has been given by God to the church. Hence, the church on earth is the authority of God. This understanding gave too much power to the popes and church officials. At one point, the church even had more power and authority than the kings of Europe.
This legacy has been diminished through the Enlightenment and certainly through post-modernism. However, this legacy still exists.
During this time there were many doctrines of the church that seeped into society on account of this idea of authority. For example, the idea of penance was birthed as well as the selling of indulgences, at which Martin Luther balked. These gave the church greater power.
Finally, the authority of the church also allowed for it to remain a constant in society, even to this day.