I'm not certain I get precisely what you're asking specifically about the Coptic church's "importance." It is an old church, notable for being a significant minority in Egypt, which is certainly an area of historical importance. It is "important" in that it is one of the Oriental Orthodox churches, with a miaphysite theology as opposed to the Chalcedonian theology of most modern mainstream churches, if that interests you at all. It is the successor to the ancient Patriarchate of Alexandria, one of the original Pentarchy of early Christianity, although there is also a Chalcedonian "Greek Orthodox Patriarch of Alexandria" who would probably claim proper succession for himself. As to its survival, Egypt was heavily Christianized (although briefly controlled by Persia) before it was conquered by the muslim Caliphate, who as a general rule were quite willing to tolerate the presence of dhimmis who submitted to muslim rule and paid the jizya, as prescribed in the Quran. There have certainly been conversions to Islam over the centuries, but the Copts continue to make up about 10% of the population of Egypt.