Capernaum occurs only in the NT, although it was perhaps the home of the prophet Nahum (the name means "village of Nahum" in Hebrew). The town is on the northern edge of the Sea of Galilee and in the time of Jesus was on the international road connecting Egypt with the Persian areas. I think that this is the reason that Jesus relocated from Nazareth to Capernaum: he wanted to be in a place where everyone came. Some of the earliest churches appear north of Israel in Syria, and I think that traders and others brought the gospel north, partly because of the work of Jesus in Galilee.
Capernaum today is not a national park, but is run by the Catholics (we saw the amusing sight of a robed monk doing a dance while his cell phone rang!). There are two main structures there that attract attention. One is a synagogue that has been partially restored. While this is not the synagogue to which Jesus would have gone, it does give a good feeling for what it would have looked like, with benches along the side and the various columns and so on. But underneath the synagogue is the foundation for the previous synagogue, which probably is the one to which Jesus went. The other structure is Peter's house. This is a house that dates back to the time of Jesus and which was expanded many times to hold large crowds. Later graffiti describes it as Peter's house. Is this really the place? Quite possible, although we do not really know. The Catholics have built a space ship church over the house. The center of the church is a glass floor through which you can look down and see the house.