Kings
After the Nephites arrived in America, up to the reign of Mosiah II (c. 592–91 BC), the Nephites were ruled by kings. Nephi's brother Jacob explains that subsequent kings bore the title "Nephi".
The people having loved Nephi exceedingly… were desirous to retain in remembrance his name. And whoso should reign in his stead were called by the people second Nephi, third Nephi, and so forth, according to the reigns of the kings; and thus they were called by the people, let them be of whatever name they would.
— Jacob 1:10–11
Judges
The last Nephite king was Mosiah II. About 91 BC, he declared that, instead of naming a new king, he would finish out his reign as king, after which the Nephites would elect judges to govern them. There were at least three levels of judges: one chief judge, several higher judges, and several lower judges. (Some passages speak of multiple "chief judges", probably synonymous with "higher judges"; for example, Alma 62:47; 3 Nephi 6:21.)
Judges were paid according to the amount of time they spent officiating. Mosiah II set the rate at one senine of gold (or the equivalent senum of silver) for one day's work (Alma 11:1, 3). He also arranged for checks in this system to avert corruption as much as possible:
And now if ye have judges, and they do not judge you according to the law which has been given, ye can cause that they may be judged of a higher judge.
If your higher judges do not judge righteous judgments, ye shall cause that a small number of your lower judges should be gathered together, and they shall judge your higher judges, according to the voice of the people.
— Mosiah 29:28–29
After announcing the governmental shift from kings to judges, Mosiah explained the principle behind the change:
The sins of many people have been caused by the iniquities of their kings....
Now it is not common that the voice of the people desireth anything contrary to that which is right; but it is common for the lesser part of the people to desire that which is not right; therefore this shall ye observe and make it your law—to do your business by the voice of the people.
— Mosiah 29:31, 26
The system of judges lasted for 120 years, when it was briefly overthrown for about three years (c. 30–33 AD) by an aristocratic cadre, led by a man named Jacob. It was replaced by a loose system of tribes and kinships, which lasted until Jesus appeared in America and established a society that approached the ideals of Zion. The society endured for about two centuries before the people fell into wickedness again.
After 4 Nephi, no mention is made of whether the Nephites used judges or kings. Mormon mentions that "the Lamanites had a king" (Mormon 2:9). His inclusion of that detail, phrased as it is, can be seen as a contrast to the Nephites having a chief judge. Since no change in government form is specifically mentioned after 4 Nephi, it is possible that the Nephites continued to use judges until their destruction in about AD 385.