There is no dog breed called the miniature collie. Most people using this term are referring to the Shetland sheepdog, a dog breed that resembles a small collie.
Shetland sheepdogs, like collies and border collies, are working dogs originally bred in the harsh environment of Scotland for the purpose of herding sheep. The border collie is believed to be the oldest of the collie types and the foundation for the other two breeds. The Shetland sheepdog was developed in the Shetland Islands off the Scottish coast and is the result of crossing local small, long-haired dogs with imported border collies. Like its collie and border collie cousins, the Shetland sheepdog is an alert, athletic, intelligent dog well suited to agility and obedience competitions.