A Mirror site is a copy of an existing site. They are used to reduce network traffic and hits to a server. Mirroring is very popular among servers that host popular large software packages for download. These servers get a lot of traffic for a single sever to support and sometimes they are forced to deny service. Having users choose from a list of mirror sites (they're encouraged to pick the one geographically closest), allows for higher download speed and continuous availability of the server.
Mirror programs are a similar concept. Several copies of the same program might exist on your disk. This is a popular technique among malware writers, as it's often difficult to catch and erase all of them.
Other examples for mirroring are disk and data mirroring, which is essentially data redundancy techniques used by large databases to ensure availability.
Hope this clears it up