The Copernican principle (in its classical form) is the principle that the Earth does not rest in a privileged or special physical position in the universe. Specifically, it derives from the claim of Nicolaus Copernicus that the Earth was not stationary, when he proposed the heliocentric model of the solar system. This had such significant implications that Copernicus himself delayed publishing the results until the end of his life, out of fear of the sort of religious backlash suffered by Galileo Galilei.
This may not sound like a particularly important principle, but it's actually very important to the history of science, because it represents a fundamental philosophical change in how intellectuals dealt with humanity's role in the universe ... at least in scientific terms.
What this basically means is that in science, you shouldn't assume that humans have a fundamentally privileged position within the universe. For example, in astronomy this generally means that all large regions of the universe should be pretty much identical to each others. (Obviously, there are some local differences, but these are just statistical variations, not fundamental differences in what the universe is like in those different places.)
However, this principle has been expanded over the years into other areas. Biology has adopted a similar viewpoint, now recognizing that the physical processes which control (and formed) humanity must be basically identical to those that are at work in all other known lifeforms.
Nicolaus Copernicus' heliocentric model of the solar system is acknowledged as the first convincing scientific demonstration that we humans are not the focal point of the cosmos.... We now realize that Copernicus' result is but one of a series of nested demotions overthrowing long-held assumptions regarding humanity's special status: we're not located at the center of the solar system, we're not located at the center of the galaxy, we're not located at the center of the universe, we're not even made of the dark ingredients constituting the vast majority of the universe's mass. Such cosmic downgrading ... exemplifies what scientists now call the Copernican principle: in the grand scheme of things, everything we know points toward human beings not occupying a privileged position.