During the sixteenth and early seventeenth centuries, there were numerous significant explorations by the Spanish explorations of North America. I will discuss a few of them below.
Perhaps the most famous Spanish explorer and conqueror in North America was Hernán Cortés. In February 1519, Cortés sailed eleven ships with about five hundred men from Cuba to the Mexican mainland. After forming alliances with several indigenous groups, Cortés led the conquest of the Mexica (Aztecs), which began the conquest of the region.
The first Spanish exploration into what would later become the United States was led by Juan Ponce de Leon in 1513. In April of that year, he landed an expedition near Melbourne Beach and St. Augustine in Florida. His mission was to search for new lands for the Spanish Empire as well to search for the fabled fountain of youth.
Hernando de Soto led a doomed expedition to North America in 1539. In May of that year, he and about six hundred men landed near Tampa Bay in Florida. Over the next three years, his expedition traveled along the north coast of the Gulf of Mexico looking for non-existent gold and getting in frequent skirmishes with Native Americans. They were the first Europeans to encounter the Mississippi River. In 1542, de Soto died of a fever. Much of his expedition had already been lost. The few survivors were eventually able to make the long overland journey back to New Spain.
Francisco Vázquez de Coronado is one of the most famous Spanish explorers of North America. Between 1540 and 1542, Coronado led an expedition of Spaniards and native Mexican allies into the lands north of New Spain. Along the way, he established a number of garrisons with the purpose of expanding Spain's influence into these lands. Coronado's expedition went into what would become the states of Arizona, New Mexico, Texas, Oklahoma, and Kansas. Along the way, Coronado got into a number of skirmishes and battles with different native groups. He was also searching for a fabled city of advanced and wealthy people known as Quivira. This fruitless search took him well into the Great Plains region before he lost faith that the civilization really existed and turned back.
One final Spanish explorer worth mentioning is Juan de Oñate. In 1598, he set out with a large contingent of Spanish settlers to settle the land in what is now New Mexico. A harsh leader, Oñate was quick to violently repress Native Americans who stood in his path of conquest. He also executed a number of Spanish settlers who tried to leave his colony and return to New Spain. In 1601, Oñate set off to finish Coronado's search for Quivira in Kansas. This, of course, ended in failure, which was made even more devastating for the Spaniard when he returned to New Mexico to find his settlement largely abandoned.