Neeeh let me specify a few details here, we're finishing this section in my history class this week... ^^
(Live in Spain, therefore, we learn a lot of spanish history xD)

*The Pizarros were brothers. One went to America first, conquered and returned to Spain, where he set out again with his brothers (and i think there were actually three...) However this first Pizarro eventually got killed by his... team, army, subordinates, whatever they were xD (anyways, there was a mutiny) and from then on it was his brothers who continued to conquer (more or less, summed up)

* The aztecs were the reigning civilization in Mexico, as they had conquered all the other previous civilizations such as the Mayas and, i believe, the Toltecs. They had a legend which told of their gods coming down to earth and teaching them many things, such as agriculture techniques, etc. These gods had white skin and hair on their faces. So when Cortés arrived, Monctezuma thought their gods had returned, and greeted them amicably. So Cortes conquered them quite easily.

* There was great rivalry to reach the Indies first, between Portugal and Spain. Vasco da Gama reached Calcuta, India in 1498, but by then Columbus had landed in the Carribean Islands and returned to Spain. Columbus was actually absolutely convinced he had reached the Indies first, as he thought the islands he had discovered were the beggining of the Islands of Japan, then know as Cipango. He died without knowing he had really discovered a new continent.
It was not until the voyages of the Conquistadores that this was discovered, specifically by an italian named Americo (name ring a bell? xD) who discovered the Pacific Ocean, though other conquistadores before him did have a pretty good idea that America wasn't Asia, he was the one who proved it

* The name of the portuguese was Enrique. Henri is the french version of that name! xP
The reigning kings in Spain were the Catholic Kings Isabel of Castilla and Fernando of Aragon, who united Spain in 1492 also after conquering Granada. Columbus's project had already been proposed to them beforehand, as well as to the portuguese king, and even the kings of France and England, but had been rejected by all. However, once the holy war in Granada was over, the queen of Castille Isabel granted Columbus another audience, this time accepting his project to reach the Indies by sailing west.

I hope that was helpful to everyone! ^^ I might update with more information later, when we actually finish the section xP
I'm kind of a history geek xD