Do Mormon's believe in the rapture of the church?

No. Members of the Church of Jesus Christ of Latter-day Saints (the "Mormon" church) do not believe in the rapture. They believe that the rapture is an unbiblical doctrine that did not exist in original New Testament Christianity.

Instead, Mormons believe that all people will go through the tribulations of the last days. At the Second Coming of Jesus Christ, those who have died will be resurrected, and all people will be judged. The righteous (regardless of their religion) will remain on the earth for the Millennium (a thousand year reign of Jesus Christ), while the wicked will be removed from the earth and endure a thousand years of torment. Following the Millennium, there will be a final judgment in which people are assigned their eternal reward (or punishment).

You can read another Mormon's response to this question at the "Related Links" below.