Excellent question! This is also a big question, so I am going to do a lot of generalizing here to keep my answer relatively concise. Hopefully, this response can springboard into useful conversations on the finer points of Spanish colonialism.

First, we have to understand that even though that map looks like uniform control over most of the Americas, in terms of actual control and settled populations, much of the Americas remained unoccupied throughout the colonial period. Case and point, though shown on this map as under Spanish dominion, Patagonia (and in reality much of the Southern Cone) was virtually devoid of Spanish settlement. In fact, this territory exercised de facto independence until Argentina’s national troops eliminated the indigenous groups who dominated the area in the 1870s. Similarly, western North America had only modest settlements and missions scattered in the vast western part of modern day United States. Spain’s control was a gradual process of colonization along extensive frontier spaces that slowly crept out from Spanish metropolises. Furthermore, the map does not show natural features like mountains, deserts, and impossibly thick forests. In these places, Spanish settlements were few and far between (if they existed at all). In my opinion, it is more useful to think of the Spanish Empire as islands of populations in a sea of unexplored and untamed territory.

In terms of physical control, Spanish explorers initially sought to control large, settled populations of indigenous people from whom they could extract natural resources. These endeavors are of course very famous. They utilized existing settlement sites, labor institutions, social structures, and cultural norms take control of the most densely populated areas of the Americas. In a similar manner in South America, semi-nomadic populations of Guaraní indigenous peoples provided a favorable environment for settlement of the Southern Cone, which led to the founding of Asunción. The city became known as the mother city of later Spanish settlements in the Paraguay/Paraná/Uruguay river basin (again demonstrating the gradual spread of Spanish settlements from a metropolis toward the frontier). These settlements were linked to one another and ultimately to Spain by trade networks and infrastructure of varying investment depending on where you were. Still Spanish control was at best modestly overseen by royal authorities and more often mostly decentralized, existing in spaces where both the “known” and the “unknown”, the “civilized” and the “untamed” existed together at the same moment.

Another key feature of Spanish control were encomiendas and missions which brought together indigenous populations into settled communities. There is a ton of research about cultural exchange, resistance, and change that occurred in these settlements, but gradually, they helped create settlement patterns that were more manageable by small groups of outsiders. Again, it is tempting to think that they represented complete control over indigenous populations; however, large numbers of indigenous people remained independent or quasi-independent for centuries.

These methods of occupation would have meant nothing if other nations had not recognized Spanish claims to the land. Spain’s claims/actions in the New World stemmed from existing European legal justification. They espoused complicated legal precedents of the right to rule newly discovered territory and Christianize the population. These were encoded in several Papal Bulls dating back from the early 1400s and had previously been used to acquire islands in the Atlantic. Spain’s claims were also bolstered by the Treaty of Tordesillas, which essentially drew a line from pole to pole, granting the territory and the right to Christianize the population west of the line to Spain and east of the line to Portugal. Again, the context of this litigation lay in complicated negotiations and power dynamics between Spain and Portugal existing shortly after the encounter with the Americas.

Over the course of centuries, these agreements changed drastically to suit the realities found in the Americas and to incorporate more international competitors. Spain did in fact face a lot of competition. Over the centuries, large amounts of their dominion fell to the Portuguese, the British, the Dutch, and the French. For example, technically, Spain controlled all of North America, but different interpretations of the legal precedent of territorial ownership led most of this territory to fall into rival hands. The Spanish monarchy based their claims of ownership on the intent to improve the land and Christianize the population; the British, for example, claimed that control of territory meant the physical occupation and improvement of the land for it to count. Therefore, territory not directly controlled by Spanish settlements could be taken and utilized. Over time, the British and French took control over these areas, which was later written in to various treaties. Additionally, foreign enemies seized Hispaniola, the Bahamas, Jamaica, Havana (for a time), and parts of northern South America. As Spanish naval power weakened, privateers attacked Spanish shipping in the Caribbean. In South America, the Portuguese continued a steady advance to the south and west along the coast toward the Río de la Plata and later inland into the heart of the continent. Spanish naval expeditions around the Southern Cone had the tacit mission to search for possible British settlements along the undefended coast of South America.

Thus, Spain haphazardly exerted their control in the Americas. Their colonization was complicated, took different forms in different places, and changed as power dynamics and economic realities developed over centuries.

Back in those days, the Catholic Church was much more powerful than it was today. Basically there was going to be a dispute between the Spanish and Portuguese Catholic monarchs over land, and to prevent warfare, the Pope came up with the Treaty of Tordesillas which gave Portugal (what is currently Brazil) and Spain everything to the West.

You are correct in that you have to break-down South America by its different geographies and ethnic groups. Peru / Ecuador etc... had the very wealthy areas due to Gold from Incas and Silver mines.

Argentina did not have a large indigenous population, and so most of the inhabitants were European settlers / ranchers (after a lot of the Indians died from disease). In fact, a lot of Argentina got its start from smuggling silver from Peru (hence the term "Rio de la Plata") and most of the power became concentrated among rich land-owners who treated the poor like dirt, and smugglers who did not care about the authority for Spain. (Or more accurately, by smuggling in through the river, they didn't have to pay taxes to Spain). This lead to current day politics in that Argentina (which was once considered to be European level living) due to its resources have fallen into hard times due to corruption. This corruption stems from historical reasons where the impunity that rich land-owners had (they basically could write their own law), and the concentration of power = corruption. In other words, compared to countries having a balanced judicial, executive, legislative, in Argentina, the rich and powerful basically owned the judicial, and could do whatever they wanted (since originally Argentina was founded by smuggling silver from Peru). If they wanted to they could make people disappear. It wasn't until their military got beaten so badly by the English that people got fed up and overthrew the military dictatorship. Lots of interesting history, so go read up on it (Each South American country is fascinating)

hi! there's room for more contributions regarding the northern reaches of Spanish colonization, but you can get a good start on this post, which contains a quick tl;dr and links to several more

• Why is Brazil the only lusophonic country in South America?

This is a large and complex question, which others have already begun to address. I want to tackle one aspect of your question, which is the idea that other nations/kingdoms/empires did not compete with Spain for American territory.

Let us dispel this notion. Both Spain and Portugal faced competition for territory and resources in the Americas. All of the great seafaring powers (England, Netherlands, France in addition to Spain and Portugal) colonized the Caribbean and the mainland, with even smaller powers controlling minor islands for various periods of time (Denmark, Sweden, Knights of Malta). Scotland has 3 notable but ultimately failed attempts to colonize the Americas before union with England.

This competition took the form of wars (all of the major powers fought each other at some point) and less overt conflict like piracy. It is also very important to keep in mind that the main theatre of war is in Europe, so while there is a armed conflict in the Americas, typically the major campaigns occur in the European theatre, and the politics and treaties of Europe included a great deal of bargaining and colony trading. If you want to focus specifically conflict in the Americas a good example for further investigation would be competition over Brazil between the Dutch, French, and Portuguese.

Now, to get to the meat of your question -- why and how did the European empires end up with their colonies in the Americas -- well it's an economic question, a social question, a political question, a geographic question, and a religious question.

The economic themes could be generalized as resource extraction (think mining, lumber/timber, furs, gems), cash crops (sugar, tobacco, coffee, cotton) subsistence farming, and smuggling/piracy. Of course slavery plays an important role here as well.

The social themes can be summed up as nobility vs crown/state interests vs farmers/peasant settlers vs merchant interests vs religious interests vs slaves vs independent indian tribes and of course the role of race.

By the political question I mean mainly relationships among the monarchies and national powers in Europe, the impact of European wars, succession struggles, civil war, control over the state and revenues, etc.

The religious question refers to the struggle between Protestant nations and Catholic nations, the role of the Catholic Church in colonization and territorial law and mediating/stoking European theatre conflict, as well as the impetus of of certain settler groups (like say, marginalized religious groups) and conversion efforts and relations among indigenous communities.

The geographic question relates to all of these, but is particular important in terms of subsistence (where can communities physically survive and not die off) economics (what can grow/be extracted where), lines of communication/travel (what can be shipped where, who can exert political control, move soldiers/troops, defend, etc), and of course the role of natural barriers in regards to settlement/frontier...dense jungle (amazon, yucatan) deserts (north american south west, parts of peru, chile, etc) inhospitable climate, (parts of canada, patagonia) mountain ranges (andes, rocky mountains, appalachian mountains brazilian highlands, mexican highlands).

I hope to go a bit more in depth about these topics when I have more time and provide some sources to investigate this. Sorry if this is a bit rambling and also subject to generalizations (it is a huge topic).