First, the root word, Millennium, is Latin for 1,000 years. In Christian theology, it refers to the first 1,000 years of Christ's reign over the Kingdom of God on earth, as described in Revelation 20:4.
A premillennialist believes that Christ will return at the beginning of the Millennium, intervening in a wicked world to then literally rule the nations for the 1,000 years as the verse states.
A postmillennialist believes that the Millennium is either 1,000 years or just a very long period of Christian governance of the world brought about by the Church advancing the Kingdom on earth—at the end of which Christ will return to a fully Christianized world. This view, popularized in the century before Darby by the Anglican theologian Daniel Whitby, became prevalent among Calvinist Puritans and their offshoots. Postmillennialism helped give rise to the idea of "the social gospel" and the belief that the Church can actually extend the reign of Christ over the earth by its actions. It has led to many churches involving themselves in politics on the premise that they are virtually obligated to lobby governments in the direction of godliness.
An amillennialist does not believe in Christ literally reigning for 1,000 years on earth but, rather, sees the Millennium as figuratively referring to the entire Church age, with Christ and the saints reigning in heaven as well as on the earth through the limited spread of Christianity. This had been a popular teaching since early medieval times, and it remains the majority position among those who identify themselves as Christians today.