South American home styles is a broad, umbrella term that encompasses pretty much every livable residence in a continent and a half, since Mexico and Central America have distinctly South American stylings but are ostensibly part of North America. Some types, like the few remaining examples of Maya and Inca architecture, you would need to bribe government with millions of dollars to own, and others, like the more modern ranch style homes you could have for a pittance, since most of their economies are less than thriving. Whatever type of Latin style you're looking for chances are if you can't find it in the Southern United States you'll can in the Caribbean or South America.
South American homes suffer from the same ripping-off phenomenon as North American homes, because the majority of people currently living there are descended from people who whose ancestors prior to the 1500s were from somewhere else. While most North Americans are descended from Europeans and thus choose to rip off Tuscan, Tudor, and Gothic styles in between trading stocks and playing basketball, South Americans are primarily descended from Spanish and Portuguese settlers, so they prefer to rip off Spanish architecture. But like salsa music is inspired by but not exactly like Spanish music, South American architecture is often inspired by but not identical to homes on the Iberian peninsula.
One popular style of home you will find all over Latin America and right up into the Southern United States is the Spanish Colonial. Adapted from the breezy clifftop villas in Spain with their Baroque stylings and ornamentation, Spanish Colonial houses feature sprawling floor plans with clay tiled roofs and arching support beams. Walls are traditionally made of smooth plaster and are usually white or off-white. Recently efforts have been made to revive the style. Much like aqua finishing solutions can revive an old painting older Spanish Colonials are being restored and new ones in the revival style are being added to match them.
The Spanish Colonial is of course not the oldest style of home in South America. That distinction belongs to what little remains of the structures built by the ancient Maya, Inca, and Aztec Empires. Of course, you'd need to be more than lucky to get your hands on one of those. Stepped pyramids with their elaborate relief carvings and hollowed-out cliff dwellings are historical sites by this point and not viable as private homes.
Modern South American architecture took a turn for the modern briefly, in contract to the fact that a lot of places still had septic tanks and such that use an industrial vacuum truck service rather than sewer systems, but have since returned to their roots. In an effort to keep their culture alive, modern South American architects are designing their buildings with ancient Aztec-inspired motifs or Spanish Colonial revival floor plans. Churches, especially are very elaborate, though they wouldn't be considered a home style unless you were a priest.