I believe there are 2 different groups of Church of Latter Day Saints. The Reorganized Church correctly points out to the schismatic Utah group that court decisions have proven their veracity. The Missouri (Reorganized) Church rejects the name Mormon, however, the larger group from Salt Lake City is frequently associated with the The Church of Jesus Christ of Latter-Day Saints and the name 'Mormon'.
I'm sure there are others on the thread better able at typing a fascinating history of their church, although discerning fact from fiction is sometimes difficult because many of that same group convolute false historical claims with truth.
The former Reorganized Church of Jesus Christ of Latter-Day Saints is now known as the Community of Christ church. Their headquarters are in Missouri. The first president of this church was Joseph Smith's son, Joseph Smith III. The presidency was handed down from father to son until W Grant McMurray was elected as president. He was the first non-Smith descendant to be the president. He resigned on 2004 after some improprieties. Their beliefs and the beliefs of the Church of Jesus Christ of Latter-Day Saints have diverged over the past 150 years. In 2000, they voted to change their name to the Community of Christ church. Here is their website Community of Christ
(Emma Smith, Joseph Smith's wife stayed in Illinois after Joseph Smith was killed. She did remarry. Most of the members of the LDS church followed Brigham Young to Utah.)
The history of the splinter groups from the Church of Jesus Christ of Latter Day Saints can be found here. Schismatic Groups The groups did start splintering off in 1831 or so, many years before Joseph Smith was assassinated.