What is the LDS meaning of the New Jerusalem?

The tenth Article of Faith, given by Joseph Smith to a newspaper as a brief overview of LDS beliefs and later accepted by the church as canonical, states:

We believe in the literal gathering of Israel and in the restoration of the Ten Tribes; that Zion (the New Jerusalem) will be built upon the American continent; that Christ will reign personally upon the earth; and, that the earth will be renewed and receive its paradisiacal glory.

(Note: The original text stated that it will be built upon "this, the American continent." The word "this" was removed after the worldwide growth of the church caused it to no longer make sense for a lot of church members.)

This article deals with LDS eschatology, and touches on both the Old and New Testaments. As the Wikipedia article notes, the New Jerusalem is mentioned in Ezekiel, and in the Book of Revelation it describes that the city would not simply be established on earth, but would descend from heaven, already built. It is frequently equated with Zion by scholars, and as the text of the tenth Article of Faith makes clear, Latter-Day Saints agree with that interpretation.

Genesis chapter 5 tells us of Enoch's surprisingly short lifespan compared to his contemporaries, cryptically stating that Enoch eventually "walked with God: and he was not; for God took him." Many Bible scholars interpret this as meaning that he was taken up to heaven without tasting death, much like Elijah. The Latter-Day Saints accept this interpretation, but also believe that Enoch was not alone.

Moses chapter 7, (part of an excerpt of Joseph Smith's unfinished project to produce an inspired translation of the Bible that would restore truths that had been lost and correct the errors of history,) tells of Enoch's work and ministry, describing him as a great prophet and a leader of men. He established a city which was named after the concept that it embodied:

18 And the Lord called his people Zion, because they were of one heart and one mind, and dwelt in righteousness; and there was no poor among them.

19 And Enoch continued his preaching in righteousness unto the people of God. And it came to pass in his days, that he built a city that was called the City of Holiness, even Zion.

The chapter closes with a description of the city's fate, just before the Flood:

68 And all the days of Zion, in the days of Enoch, were three hundred and sixty-five years.

69 And Enoch and all his people walked with God, and he dwelt in the midst of Zion; and it came to pass that Zion was not, for God received it up into his own bosom; and from thence went forth the saying, Zion is Fled.

Joseph Smith taught that the location of Zion, the city of Enoch, had been upon the American continent, and that it was to return from heaven as the prophesied New Jerusalem at the beginning of Christ's millennial rein, and be established upon its original site, in modern-day Missouri, and be joined with the righteous community that was established there in modern times. He explained that this was an important part of the meaning behind Isaiah 2: 3 (emphasis added):

3 And many people shall go and say, Come ye, and let us go up to the mountain of the Lord, to the house of the God of Jacob; and he will teach us of his ways, and we will walk in his paths: for out of Zion shall go forth the law, and the word of the Lord from Jerusalem.

Joseph Smith taught that the last part of this verse was not simply parallelism; it literally referred to two distinct places, and during the Millennium, there would be two world capitols, a political capitol at Zion (the New Jerusalem) and a spiritual center in Jerusalem.

For a more in-depth treatment of the topic, see Talmage's "The Articles of Faith", lecture 20 (covering Zion) and lecture 21 (covering Christ's millennial reign.)