It was Christopher Columbus who first arrived, and once the link had been established between the continents the Spanish and Portuguese came to an agreement to split the new territories equally.
It was only in the following century that other countries – such as Britain, the Netherlands and France – got involved with the division of the land. The objective of the discovery was to find a better route to India and China to improve their trade routes. It was between 1492 and 1503 that Columbus was confirmed as finding the new set of continents, though it wasn't followed up with any other conquests for over 10 years in 1517.
The first location that was taken after the long break in expeditions was Hispaniola, and the areas of land were broken up into tow, for Mexico was to be taken by Hernán Cortés and the Inca by Francisco Pizarro. The rate at which the population of the indigenous people fell was astounding: 80% of the indigenous population in these areas had disappeared in just 150 years. This gives an idea of how much the European interference had an effect on the new world. Reasons for the drop include the harsh treatment of the locals and the introduction of new diseases brought from Europe. There was much debate regarding the locals, much of which focused on their religious beliefs.
The religious aspects of the European colonization of Latin America took on new ground in 1524, when attempts were launched to Christianize the locals. It was during this time period that much conflict arose and sadly many natives were killed for refusing to convert. As many were against the idea of conversion it pushed many of the locals away from any kind of cooperation, which, in turn, caused many internal conflicts for the Spanish regarding the direction of how to appropriately deal with their actions. Having said this, it wasn't all bad for the Spaniards, as their financial success was evident.
A significant proportion of their affluence came from gold and other valuable items taken during the course of the 16th century. Although the slavery of the local people had reduced, they then used African slaves to compensate. Later years for the European colonization saw several diseases that tore through the land and wiped out much of the population.