The major effects Copernicus caused were in religion. Suddenly the theological perspective of man's dominance in a dominant world was shown to be no longer tenable. Though Pope Paul III was not an outspoken opponent of Copernicus, his bishops and cardinals were. On their behest, the Church officially condemned Capernicus's cosmology. When friar Giordano Bruno, himself a mathematician and astronomer, upheld Copernican heliocentricity, he was subjected to execution at the order of the Inquisition, in place to find and destroy heresy. Heliocentricity uprooted the theological doctrine that the universe revolved around Man as the paramount and most glorious end-result of Creation, with the Sun, the Stars, the other Planets and the whole Universe revolving in an Aristotelian pure circle around Man, dwelling upon Earth.

I would say that one of the major impacts we see of this today is our increasing trend towards secularism.  I am not saying that Copernicus on his own caused this.  However, his theory helped.  The reason for this is that it helped to get people to challenge the orthodoxy of the Church.  As the Church lost its place as arbiter of scientific truth, it lost some of its overall authority.  At the same time, people were being encouraged to think more scientifically, which can also lead to less religiosity.

Therefore, I would argue that Copernicus helped to bring about our modern society that gives less weight to religion than previous societies did.

The Copernican revolution probably had its biggest impact on the way people viewed their place in the world.  The heliocentric model of the universe no longer placed man at the center of the universe and brought about quite a bit of religious upheaval for obvious reasons.  Eventually Protestants would be much more accepting of the idea than Catholics.

Some people consider the Copernican Revolution to be the start of the Scientific Revolution, so its impact in that regard in unmistakable.  As people began to look for physical evidence and experiments to explain the world around them (versus religious doctrine) a wealth of new information and ideas were uncovered.

To some degree one could also see the ideas of this revolution reflected in the cultural and philosophical shift to a more humanistic view of the world, which could easily be linked to the Copernican Revolution.