How did Portugal get Brazil, when Spain got every other South American nation?

Well, the treaty of Toredesillas in the late 15th century might be confusing today, but you have to remember...

At the dawn of the 16th century, Europe was nothing like it would be just 100 years later with the 30 year's war. England was still a hopelessly poor nation (they having not discovered the new world or trading yet.) France was fairly strong, but just coming out of the 100 year's war, they still had to deal with a far more pressing matter. Whilst the city states of Venice, Malta, Lombardy... Were all thriving with culture, France had to work out their national identity. The complete but-kicking that they got during the 100 year's war, such as at the Battle of Agincourt, caused them to realize that they were decades behind other nations- not just in military tactics, but much much more. Although this is a theory I would love to get into more, in short, a big problem during the 100 year's war was the fact that the petty dukes of France refused to listen to the king, or only did whilst trying to curry flavor. The dukes often fought over who got to go and who didn't get to go to war. Also in England, although they weren't Anglican yet, the church had less of an influence in the daily life of the English court, whilst in France the State wouldn't be secularized until the French Revolution. That leaves only a few other superpowers. Russia was still a backwards nation devoted to farming, and still having to deal with the relatively strong Golden Horde Khanate centered around Crimea. Poland was relatively strong, but until the Polish-Teutonic war in the early 16th century, they would never be the dominant power. Italy was still a shattered region, with only Venice being relatively strong, and not even close to its apex as a power. That only left three choices. The Ottomans were far too busy in the 15th-16th century, heck, even until the 18th century, to try to become a naval power. Although their navy was strong, at least compared to Russia in the early wars of the 1800s (The only time the Ottoman naval powers would ever be tested seriously), their army was far too busy carving deeper and deeper into Austria, almost to the gates of Vienna itself. That leaves only two choices: Spain and Portugal. Spain was obvious. After driving the Muslims out of the majority of Spain, to the Gibraltar peninsula itself, Spain was very strong, and at a culturally defining moment. In fact, historians call this moment the "Reconquista" era, because it was such a defining era for the Spanish. Whilst our mutual friend of Portugal (whom we will talk about later) was busy sailing around the cape of Africa to India, the Spanish truly were looking for new ways to expand their empire. This is why Queen Izabella sent Chris Colombo across the world- to expand their empire to the rich "East" Indies. When Spain found a new continent, although we think of European powers as extremely powerful (which only comes later), they were truly one of the only powers who could conquer it. Unlike the French army, the Spanish one was extremely well equipped with state of the arms weapons, especially the new guns they had. This allowed them to quickly sweep over the natives. When Portugal, the only other sea-faring power of Europe wanted in, it was serious. Although it's easy to think of Portugal as Europe's Detroit (along with Greece) today, in fact, Portugal was easily the most dominant naval power in the 15th century. They discovered the way around Africa, a sea-route to India, and perhaps even Australia. The answer can relatively simply come down to this. The pope, who was a political figure, even more so than today, wanted to curry flavor with both the Portuguese and the Spanish. So, in the Treaty of Tordesillas, he divided the New World in "half". Maps back then displayed the western part of the New World as very thin, perhaps only an Island chain. When it was discovered not to be so, the treaty simply, and unofficially, dissolved. England took the north, Portugal expanded their borders west a bit, and Spain consolidated their hold.

TL;DR: THE POPE LOOKED AT THE WRONG MAP; AND NO ONE WAS POWERFUL ENOUGH TO STOP HIM