There have been lots of wars in South America. This is not a complete list, but these are some that come to mind in the Southern Cone:

• Various Colonial Revolts--While they were ruled by Spain, there were a number of significant revolts including the Guaraní Uprisings, Revolt of the Comuneros, and the Túpac Amaru Rebellion.
• Various Conflicts between Portugal and Spain over the Río de la Plata--This area of South America was where the frontiers between Spain and Portugal came together, resulting in frequent fighting and negotiations between the powers. Portugal wanted to tap into the contraband trade that flourished in the Banda Oriental. These conflicts ultimately led to the expansion of Portuguese territory far beyond the original line established in the Treaty of Tordesillas.
• British Invasion of Buenos Aires--In 1806 and again in 1807, British forces invaded Buenos Aires. The Spanish leaders fled, but local creoles led successful defenses of the region and ultimately drove out the British. The conflict led to the growth in power of creoles in the region, gave rioplatense leaders a taste of free trade and self-rule, and helped develop a sense of identity that hadn’t existed before.
• Independence Wars--These were prolonged struggles that took place all over the continent for more than a decade. Sparked by Napoleon’s invasion of Spain that created a power vacuum, the most famous South American leaders like Bolívar, San Martín, and O’Higgins drove out the Spanish and began the lengthy process of state-formation. They established some of the earliest republics in the world, though they were deeply flawed institutions. This resulted in...
• Various Civil Wars--Throughout the nineteenth century, South American countries were beset by internal conflicts as they experimented and debated democracy, issues of inclusion, and the balance of power in the new nations. Some of these include the Uruguayan Civil War, the Argentine Civil Wars, and the Chilean Civil War. There were lots of others during (and after) the state-formation period.
• Cisplatine War--This was a conflict that broke out between Buenos Aires and Brazil over the Banda Oriental in 1825. The conflict was largely a stalemate, which ended when Britain led negotiations and helped form the independent country of Uruguay.
• Paraguayan War--This was a conflict between the Triple Alliance (Brazil, Argentina, and Uruguay) and Paraguay from 1864 to 1870. It was one of the bloodiest wars of the nineteenth century, yet very few people know anything about it. Check out this nice article from the Economist for a summary of the conflict. It resulted in a crushing victory for the Allies. Paraguay lost roughly 40% of its population and 25% of its territory and, consequently, remains a deeply impoverished place.
• Chincha Islands War--From 1865 to 1866, there was a small naval war fought between Spain and an alliance of Peru and Chile. Bolivia and Ecuador later joined the conflict as well. Spain seized the Chincha Islands, which had spectacular deposits of guano, because Peru refused to pay indemnities for their Wars of Independence. A Spanish squadron proceeded to raid shipping and blockade ports along the Pacific coast. However, the squadron did not have access to land forces, so there was very little they could do. Eventually, the combatants negotiated a peace treaty which recognized Peru’s independence and returned the islands.
• War of the Pacific--War broke out between Chile and an alliance between Bolivia and Peru over the Atacama Desert, which had huge deposits of valuable nitrates. Chile won the war thanks in part to superior leadership and control of the seas. Bolivia lost access to the Pacific Ocean, becoming a landlocked country.
• Conquista del desierto--I’m using this as a collective term for the final conquest and subjugation during the 1870s and 1880s of independent indigenous groups in Patagonia and Araucania who had lived in virtual independence since Spaniards arrived. The consolidated nations of Argentina and Chile undertook these campaigns of extermination to put an end to the threat of raids and to open up their respective frontiers to settlement. Both nations won resounding victories, though today the legacies of these conflicts have become intertwined with accusations of genocide.
• Chaco War--This war was fought between Bolivia and Paraguay from 1932 to 1935 over the Chaco region in central South America. It was the bloodiest conflict in South America during the twentieth century. It gained the nickname “the green hell” because conditions for soldiers were so deplorable. The conflict originated over a border dispute in the sparsely populated region in which both countries sought to improve their regional standings after previous defeats in nineteenth century wars and gain access to resources. Paraguay won and annexed the Chaco, giving it the shape it has today.
• The World Wars--Brazil fought in both World Wars. I don’t know nearly enough about Brazil to expand on that topic here, but you might be able to find past discussions on the topic on AskHistorians.