Similar to Oceania and S.E. Asia, South America had small kingdoms and countries established, but not much. The two early kingdoms were the Chimu and Inca.
The Incas were the largest and strongest empire before the Europeans came. They were established in modern day, Peru. Machu Picchu is the only Incan city that survived the Spaniard invasion and is popular tourist attraction to this day.
After the journey of Christopher Columbus in 1492, Europeans started colonizing in America. The Spaniards and Portuguese colonized much of South America, although France, Dutch, and Britain did colonize a little bit, but it was mostly colonized by the Spaniards and Portuguese.
Mexico was colonized by the Spaniards, and was home to two different colonies, the Aztec and Mayas. By 1750, the Spaniards expanded to a little bit of North America including much of Florida, and the Caribbean.
The Europeans expanded to the Caribbean and used it for slave trade. This continued till the 19th century before it was abolished. Some islands were overrun by the slaves and became independent of Europe. Some were allowed to be separate countries, and others still owe allegiance to Europe.