Which countries colonized various regions in the New World?

The Spanish colonized what we now think of as the Southwestern United States, plus Florida, as well as Mexico and much of Central and Latin America, as well as Cuba. The Spanish explorers or "conquistadors" responsible for colonizing these areas were Christopher Columbus, Hernan Cortez and Francisco Pizarro, to name the most prominent among them. 

The Portuguese colonized Brazil, building enormous system of plantations there. The Dutch first settled what is today New York and New Jersey, which they named New Netherlands, and which they farmed very successfully for a time. The English settled what is today Virginia, and then Massachusetts, Road Island, Pennsylvania, Maryland, Vermont, the Carolinas, Georgia, Delaware and Connecticut. In the mid-1600s, the English fought and defeated the Dutch, and took New York and New Jersey for themselves.

In Canada, the French Jesuits settled what became Quebec. Both the British and the French also made early forays into the settling the Caribbean islands as plantations.