I will try to keep this as simple as possible. Please post a comment or question if you want anything answered or discussed more in depth! (Also feel free to correct me if I have made any errors in discussing Catholicism or Protestantism as I do not wish to misrepresent their histories).
I will briefly discuss the founding of the Catholic, Protestant, and Mormon churches.
CatholicThe Catholic church claims to have received its authority from succession going all the way back to Peter the apostle. No Christians dispute the authority that Christ gave to his chief apostle, Peter. The Catholic church claims that Peter past this authority on and a couple hundred years later the Roman Catholic Church was founded.
ProtestantIn the early 16th Century reformers became disenchanted with the apparent corruption in the leadership of the Catholic church, including the selling of indulgences and worship of saints and images. Martin Luther was one of the earliest reformers. Martin Luther was excommunicated from the Catholic church for believing that ultimate authority came from the Bible and not from the Pope. He later founded the Lutheran church. Zwingli and Calvin were two other influential protestant reformers. Basically the protestant faith stemmed from Catholic church and kept some Catholic beliefs while also developing new beliefs based on the Bible. Their claim to authority comes from the scriptures.
Latter-day Saint (Mormon)The LDS (Mormon) Church is not a Protestant church because it did not stem from the Catholic church or another Christian church. In fact, it is the
only Christian church that is not Catholic or Protestant. In The Church of Jesus Christ of Latter-day Saints was founded on April 6, 1830 by a latter-day prophet, Joseph Smith.
As a young boy, Joseph was confused by the religious contention that surrounded him in New York during the Second Great Awakening. He wanted to know which church to join and was having difficulty resolving this question using the Bible. One night he read this verse in the New Testament:
If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him.
James 1:5
This scripture convinced him that he must pray to God and ask for the wisdom that he lacked (which church to join). He went to the woods, prayed, and received a miraculous response. God the Father and his Son, Jesus Christ appeared to the boy Joseph and answered his question, that he should join none of them. Joseph was chosen to be a prophet in these last days.
Through this prophet, God restored his gospel in its fullness, including the Priesthood (the same authority Christ gave to Peter and the other apostles). The same church organization, structure, and doctrine that Christ established over 2000 years ago was restored to the earth with Prophets, Apostles, Evangelists, Bishops, etc. Just as every prophet that lived before Joseph Smith, the Lord brought forth new scripture for our day, including the Book of Mormon, Doctrine and Covenants, and the Pearl of Great Price.
Currently, Thomas S. Monson is the living prophet on the earth. The authority that God restored to the Prophet Joseph Smith has been passed down through a succession of prophets and continues with the current prophet. There is also currently a Quorum of Twelve Apostles and the same church organization that Christ had when he organized his former-day church, the Church of Jesus Christ.
ConclusionI hope that the differences between the churches is apparent, through their histories. I went into more detail about the Mormon church because I know more about its history. The Catholic church is supposed to be a continuation of the original Christian church, the Protestant faith branched from the Catholic church a few hundred years ago, and the Mormon church is a restoration of the original Christian church after an apostacy (falling away) from the truth.