The two empires that were eventually defeated by the Spanish were the Incan Empire, which occupied what is now Peru, and the Aztec Empire, which comprised much of what is now Mexico. I would not include the Mayan Empire here because, by the time the Spanish arrived, the civilization had already been in decline. Formerly great cities were, by then, buried under rain forest.
The Spanish assumed control over their colonial subjects with a combination of guns and germs. Their weaponry, wielded by relatively small armies, allowed them to take over cities. Their "germs," including small pox, influenza, and yellow fever, decimated native populations.
Hernan Cortes sailed into Mexico in 1519 with eleven ships, a little more than 500 soldiers, and 100 sailors. He also brought horses, which were introduced into the New World by the Spanish.
Cortes's decisive victory over the Aztecs occurred in the famous city of Tenochtitlan. In his first battle there with the Aztecs, the Spanish lost. However, Cortes returned in December 1520 and conquered the capital city street by street. As a result, by 1521, the Aztec Empire had fallen. Cortes then went on to Central America, particularly the Honduras, to seek new territories for Spain.
Francisco Pizarro conquered the Incan Empire. He, too, brought horses, but far fewer men -- only 180. He also only had one ship. On the other hand, the Incan emperor had an army of 30,000 men.
After a period of imprisonment, imposed as a result of his demand that the emperor and his subjects accept Christianity, Pizarro ordered an attack on the nation, later seized the emperor Atahuallpa himself, and had him killed by strangulation. The Incan army retreated, probably intimidated by Pizarro's artillery, allowing the Spanish to seize the empire in 1533.
After returning to Peru from a disappointing trip to Chile, which he had intended for future conquest, Pizarro was re-imprisoned -- this time by his brother, Hernando -- and executed.