Mosiah
There are two individuals named Mosiah in The Book of Mormon. They were grandfather and grandson, respectively, and both served as king of the Nephites at Zarahemla.
Mosiah, the granfather, was a Nephite prophet who led the Nephites from the land of Nephi to the land of Zarahemla and was later appointed king. He was the father of King Benjamin. Mosiah, the grandson, was Benjamin's son and was king of the Nephite nation from about 124 BC to 91 BC. The Book of Mosiah is named after Mosiah, the grandson.
Mosiah, the grandfather
The history of the first Mosiah is limited to Amaleki's account in the Book of Omni (Omni 1:12-23).
Following a period of "much war and contention between...the Nephites, and the Lamanites" (Omni 1:10), Mosiah1 was "warned of the Lord that he should flee out of the land of Nephi, and as many as would hearken unto the voice of the Lord should also depart out of the land with him, into the wilderness" (Omni 1:12). The journey led them to the land of Zarahemla, inhabited by a group of people who had come from Jerusalem at the "time that Zedekiah, king of Judah, was carried away captive into Babylon" (Omni 1:15). The two groups united and appointed Mosiah their king.
Mosiah, the grandfather, also translated engravings found on a stone which gave an account of the Jaredites, another people who had previously inhabited the area.
Mosiah, the grandson
Mosiah instituted a new governing system on his deathbed after all of his sons refused to succeed him. After Mosiah's death in approximately 91 BC, a council of elected judges governed the land until Christ appeared (see Mosiah 29).