Conquest of Mexico and Peru
The first step towards the conquest of America was taken by Hernan Cortes, who in 1519 was designated by Diego de Velasquez (governor of Cuba) to take an expedition and head to lands that had previously been visited by Francisco Hernandez de Cordoba and Juan de Grijalva in 1517, who had heard of the existence of a rich empire (which was none other than the Aztec empire).
This way, on November 8th, 1519, Hernan Cortes arrived to Tenochtitlan, the capital of Mexicas, where we was received by tlatoanis Moctezuma II. After a couple years of fighting, the conqueror achieved Aztec surrender (August 13th, 1521). Because of this, the empire fell into the hand of the Spanish and would later become New Spain, the first American viceroyalty.
Continuing with their conquest, the Spanish took control of Panamanian territory and heard a rumor that towards the south they would find extensive lands rich in gold. This is how, in the year 1524, Francisco Pizarro, Diego de Almagro and cleric Hernando de Luque joined in their intention of conquering said land.
With royal permission, they embarked towards the coasts of the south in 1531 and arrived to Tumbes (north of Peru). There, they heard of the war that faced off the brothers Atahualpa and Huascar for the succession to the throne of their deceased father, Inca Huayna Capac.
The Spanish managed to meet with Atahualpa (who had already defeated Huascar). After a few conversations, an armed confrontation took place between both sides, which ended with the Inca’s capture.
During his stay as a prisoner, the emperor perceived the obstinate desire on the part of the Spanish to attain wealth, and this way, taking advantage of their weakness, decided to offer a shipment of gold in exchange for his freedom. Pizarro accepted the proposition and Atahualpa sent for the treasure. However, the conqueror decided to not hold his part of the deal and killed Atahualpa (July 26th, 1533).
With the Inca dead, Pizarro took Cuzco in 1554, and he founded Quito and Trujillo that same year. Then, in 1535, he founded the city that is nowadays known as Lima, settling Spanish power in the Inca empire for good.