Why is it called The Church of Jesus Christ of Latter-day Saints?

The Church is so named because:

1. Mormons believe that this church is the same church that Jesus Christ established when he was on the earth, same organization and everything. It's his church and his doctrine. Therefore, the Church should bear his name.
2. Mormons believe that after Jesus died, the Church began to lose the purity of the teachings of Jesus Christ (the Gospel). The Church started teaching the philosophies of men mixed with the Gospel, and so it stopped being God's true church. Christianity became divided and corrupt, and it remained in this confused state until the world was ready to receive the pure Gospel again. In New York in 1830, Jesus Christ restored his church through the prophet Joseph Smith. (See Joseph Smith (mormon.org)) Joseph received a revelation that God wanted the name of the church to be the Church of Jesus Christ of Latter-day Saints. So that's what he called it. 
3. Members of the Church are called saints because that's what members of the ancient church were called. Members today are called latter-day saints because we are living in the Latter Days, the period of time that takes place just before Jesus' second coming.
4. The Church is often referred to as the Mormon Church, and members are commonly called Mormons, because they believe in the Book of Mormon as well as in the Holy Bible. The Book of Mormon is another witness of Jesus Christ. It recounts the history of two ancient-American nations, the Nephites and the Lamanites. The Book testifies of Jesus Christ and teaches his Gospel. (See The Book of Mormon (mormon.org))

The Church could have been given this name: The Church That's the Same as Christ's Ancient Church Whose Members Are Living in the Time Just before the Savior Comes Again. After that, the actual name of the church doesn't seem too long, now does it?

See also these Mormon.org articles: The Restoration of Jesus Christ's Church, Christ's Church

What do Mormons think of Joseph Smith?

Joseph Smith was a poor, hardly educated farm boy from New York who was trying to find out which church he should join. In 1820, when he was 14, he went into a grove of trees on his family's farm so he could have some privacy, and there he prayed out loud for the first time by himself. He wanted to know which church he should join because there was a lot of religious upheaval where he lived - some joining this church, some joining that church, and the leaders of the different sects were preaching different things because they were interpreting verses from the Bible differently.

When Joseph prayed, Heavenly Father and his son, Jesus Christ, appeared to him. They told him not to join any of the sects, for they were all wrong, meaning that none of them was Their true church. They also talked to Joseph about other things, but Joseph didn't record what They said.

This experience, known as the First Vision, is extremely important to members of the Church of Jesus Christ of Latter-day Saints, and to the world. It teaches us that Heavenly Father and Jesus and two physically separate beings, though they are one and united in basically every other way. They even look alike. It also means that Joseph was about to become an instrument in God's hands to restore God's true church, what is now the Church of Jesus Christ of Latter-day Saints, to the earth.

Joseph also translated the Book of Mormon from a set of gold plates that was given to him by an angel of the Lord. It is an ancient record of the inhabitants of the Americas from the time of the Tower of Babel down to the destruction of the Nephites.

Joseph Smith suffered lots of persecution. If you were Satan, and God was trying to restore His church, wouldn't you want to do all you could to stop it? So, Joseph Smith and fellow church members suffered all kinds of attacks, violence, and injustice. They were forced by this persecution from Ohio to Missouri to Illinois, and then to Utah where there remains a large Mormon population today. Joseph didn't make it to Utah, though, for he was killed by a mob at Carthage, Illinois. In spite of its rocky start, the Church continued to grow and is today a worldwide church.

Mormons don't worship Joseph Smith, but we revere him as a great man who had to go through hard things to serve God and the rest of us. It was a tough job, restoring the Church and translating the Book of Mormon and taking care of the affairs of the new organization. It shouldn't be surprising that we Mormons love Joseph Smith as much as we do.