mostly copying from my own response about how U.S. President Rutherford B. Hayes sort of saved Paraguay from being wiped of the map:

In late 1864, Paraguay, under President Francisco Solano Lopez, invaded both Brazil and later Argentina (in spring 1865), on the pretext of supporting the Uruguayan Blanco faction in their ongoing civil war against the Colorado faction. The Blancos were not faring well against the Colorados, in large part because the Colorados were supported by both Argentina and Brazil (who had their own rivalry but found themselves on the same side here). The Brazilians and Colorados concluded their victory in Uruguay by February 1865, and thus the Triple Alliance was born. Uruguay, Argentina, and Brazil signed a treaty guaranteeing that all three would fight until Solano was overthrown and their territories restored.

The Paraguayan army won some early victories and took over territories that had long been disputed- but that were sparsely populated, mostly by indigenous peoples. Solano's army was quite large, especially respective to Paraguay's small size and population, but not particularly well equipped or trained. Solano, of course, had either severely over-estimated his own forces or had underestimated his foes' wills to fight over relatively unimportant territories. Once the Triple Alliance was properly mobilized, it was all but a foregone conclusion that Paraguay would lose the war, as they were severely outnumbered and outgunned. The naval battle of Riachuelo (1865) on the Parana River and the battle of Tuyuti (1866) were both crushing defeats for Paraguay, but Solano kept fighting for years after, even undertaking a guerrilla campaign after the capture of his capital of Asuncion. This guerrilla war caused an incredible death toll among his own population, but was unable to save Solano's regime. In 1870, Solano was killed and the war came to an end.

The war had been beyond devastating to Paraguay- there are extraordinarily wide estimates on the human cost of the war (due to the unreliable census records) with some estimates surpassing half of the population dying. It was likely much lower than 50%, but suffice it to say that a horrifying portion of the population was killed or died of hunger and disease during the war, including most males of military age. Treaties were signed by Paraguay that confirmed that disputed territories in modern day Mato Grosso do Sul were Brazilian, and that the present day provinces of Misiones and Formosa belonged to Argentina. The issue of what to do with the Chaco Boreal, a sparsely populated region disputed between Paraguay, Argentina, and Bolivia, was left unresolved until 1878. Argentina and Paraguay agreed to submit the issue to a third party, in this case the United States of America. The Chaco Boreal represented about half of Paraguay's remaining territory- so its loss would have been another insult on top of many injuries suffered in the war. But President Hayes chose to allow Paraguay to keep the territory, and for this he ended up with a city, a province, and various other small things named after him in Paraguay.