Someone else has already provided an answer, which I consider to be rather one-sided and incomplete in several aspects, so I'll try to provide more ample context.
Economists will tell you "it's the economy, stupid!". Political scientists will convince you that it was policy making. I will make the case that it's far more complex than that, because I'm a historian, everything is complex. So come fly with me, let's float down to Argentina.
First and foremost, Argentina was never the world's richest country. It was close, but it never quite made it there.
The period of fastest economic growth was achieved by an agro-export model, which essentially meant economic development via the elevated prices of agricultural and cattle derived resources and products. See, Argentina had only become a unified country after the 1853 Constitution was signed and ratified by Buenos Aires province in 1860 (which they only did after being granted several considerations, especially one that forced the other provinces to pay five years’ worth of war budget to Buenos Aires, which was an enormous amount of money, but I digress). The next decade saw the consolidation of centralized power in the hands of an oligarchical elite that saw great potential in expanding the borders to the South, into Patagonia, populated by several native peoples who they deemed to be barbaric savages.
In 1874, President Nicolás Avellaneda was sworn in. He had intended, following in the footsteps of his predecessor Domingo Faustino Sarmiento, to induce an influx of European immigrants that could work the land. In 1876, he introduced to Congress the Law of Immigration and Colonization N°817, which sought to promote Argentina as a growing economy, making it attractive for immigrants, who would be granted land for farming and cattle raising, while also authorizing the creation of exploratory expeditions into the “uninhabited” areas south of the border.
Heavily influenced by the Eurocentric beliefs of the civilizing mission and the American manifest destiny, the oligarchy used several native malones, raid parties the natives did to steal cattle, as the perfect excuse to exterminate the natives in what is now called the Conquest of the Desert, a series of military campaigns deep into native territories led first by Adolfo Alsina, then Minister of War, and second by general Julio Argentino Roca. The ensuing genocide saw thousands of dead and tens of thousands of displaced natives, and ensured the acquisition of hundreds of thousands of square kilometers.
Roca would go on to become president in 1880, starting what we now call the 80s Generation, a series of oligarchical governments that used electoral fraud to keep themselves in power for decades, up until 1916. During this period, their economic policy consisted of the aforementioned agro-export model, allowing the Argentine economy to grow exponentially, all the while increasing our dependency in foreign manufactured imports. This dependent model would go on for decades, during which time finances bloomed, allowing Argentina to have its very own belle époque. This meant riches beyond their wildest dreams for the upper echelons of society, and increasingly suffocating impoverishing circumstances for the working class, who did not share in the bounty.
The height of this dependency was achieved in the Roca-Runcimann Pact, a trade agreement between Argentina and the United Kingdom signed in 1933, under the presidency of Agustín Justo. The deal guaranteed the continuing export of meat (almost solely processed by Argentina based, UK owned frigorific companies), and the continuing of coal import. Argentina agreed to only buy coal from the UK, tariff free. Keep in mind that this predatory deal was brokered in the aftermath of the 1929 crash; Argentina only produced meat and agricultural products, lacked almost any industrialization, and needed somewhere to sell meat, during a time in which most nations had decided to stop depending on foreign exports for meat consumption.
Even though the elite continued to reap a plentiful harvest for themselves, Argentina’s economy was no longer booming. Political instability caused several military coups, including the one that deposed Juan Domingo Perón, of whom I’ve spoken before hereand here, and whose economic model consisted of the substitution of imports by industrializing the country. While I was writing this, I saw someone else point out that the main cause of our economy’s “downfall” was Perón and his populist legacy, a generally conservative diatribe stemming from a rejection towards policymaking considered to be "populist". Perón's pseudo Keynesian economic model of welfare State aimed to industrialize as a means to stop Argentine dependency in foreign products. This model did not account for several factors, including a progressive and sustained increase in public spending by subsidizing the private sector and social benefits, and the impossibility of achieving self-sufficiency by Argentina's limited natural resources and the inability this caused to produce every single item required for a society to grow economically. This factors, among others, led to an increase in the conditions that would eventually lead to recessions.
I would also like to talk about other people who were considerably worse for Argentina’s economic development in the later decades of the century.
José Martínez de Hoz
Martínez de Hoz was the heir to a dynasty of oligarchical landowners, and he served as Minister of Economy for two dictators during the last military dictatorship, of which I spoke here. As an oligarch, his economic policy was to be expected: he began tearing national industries apart, while taking gargantuan loans, freezing wages, eliminating the inheritance tax, favoring lobbyists and facilitating tax evasion by Argentina’s largest corporations. His policies caused inflation to skyrocket and the peso to depreciate exponentially which, paired with devaluated wages, led millions of people into poverty.
When the military junta lost the Falklands War, they were forced to take a more moderate approach to governability, placing the next of our personages in center stage, as the president of the Central Bank.
Domingo Cavallo
Following Martínez de Hoz’s tendency to favor lobbyists, Cavallo allowed privately owned corporate debt to translate into State liabilities. When the dictatorship ended and Raúl Alfonsín was democratically elected in 1983, Cavallo left Buenos Aires in order to isolate himself from scrutiny, which would soon come, because the disastrous policies implemented by the dictatorship left the newly recovered democracy with a hyperinflation and a nearly worthless currency.
When Alfonsín saw himself forced to leave power early due to inflation having reached a staggering 200% a month in 1989, Carlos Menem became president, with Cavallo influencing many economic decisions, and eventually becoming Minister of Economy in 1991. His response to the crisis? Why, to change the currency, of course. Argentina left behind its older currency and replaced it by the current version of the peso, which, during what was known as the convertibility plan, was worth one dollar. Cavallo devised the plan to keep the exchange rate at 1:1 in order to reduce inflation and promote international inversion in Argentina. A new belle époque was here, thanks to neoliberalism, that was marked by the privatization of nearly every State-held company and industry, chief examples being Aerolíneas Argentinas, our national airline, and YPF, our national oil and gas company, and the complete and absolute dismantling of the railway system, which spanned tens of thousands of kilometers.
Recession became inevitable, and the turn of the century saw millions of unemployed, homelessness and crime rates higher than ever before, and the complete collapse of the unsustainable convertibility plan. However, the subsequent government, led by Fernando de la Rúa, called Cavallo to head the Economy Ministry once again, to try and salvage the deep recession Argentina was in during 2001. It was too late, his past decisions had caused the country risk to rise, causing the IMF to refuse a renegotiation of our external debt, and the biggest capital flight and bank run in our history. The economy collapsed, leading Cavallo to take desperate measures to stop the cash flow, imposing what we call corralito, or financial corral, which impeded citizens from withdrawing money from their bank accounts. Following civil unrest, a state of emergency, several deaths caused by police repression, and dozens of massive protests and demonstrations, De la Rúa was forced to resign the presidency.
Bibliography
Edit: some clarifications.
The eternal question.
To start, it is worth putting one thing right: Argentina is much, much richer today as in 1890. Their RGDP/capita in 1890 is $4,245 (2011 $GK) and in 2016 it was $18,875. So, real income per person is about 4.5 times higher today than it was then. Of course, as jumped-up plains apes (as Brad DeLong says) we are obsessive about relative wealth, so we remember the 1890s as a time when Argentina was rich, and today we think of Argentines as poor (though they are still well, well above the global average GDP/cap!) But the actual living standards of modern Argentines, after 130 years of "failure" is vastly higher than it was back then.
But still, as a % of US real GDP, Argentina has fallen from about 85% of US GDP to about 30%, so that's a pretty big relative decline. There is obviously something to explain here.
We should also look at things things in regional context. Argentina's economic performance is only slightly worse than the rest of Latin America, and most of that is down to how far head it was in the early 1890s. Uruguay, right next door with very similar economic and demographic structure, performs almost exactly the same. The overall pattern is not so far out of line with the rest of Latin America, though always starting higher and usually growing a bit slower. So, Argentina is not really an exceptional case, except insofar as there was this brief window in which people thought it ought to be, which is exactly the end of the 19th century.
It is also worth putting things in temporal context. The late 19th C. was a time of historically high terms of trade for primary producers. Prices for foodstuffs and the inputs into industrial processes were high, and the prices of manufactured goods had been driven down by industrialization in Europe and parts of North America. Those places with factor endowments that allowed them to produce these goods and export them, saw their incomes increase enormously. For Argentina, this meant cattle products (hides and meat), wool and wheat. (Argentina is not even the most extreme example - just look at the unbelievable boom and bust of the Manaus rubber boom, or the collapse of the guano and nitrates industry on the West Coast!) Other countries that share this quality - Uruguay, but also Canada, Australia, and the US midwest - also do extremely well in this period.
This means that GDP per capita as measured in those places was very high, but it is worth remembering that quite a lot of this is a price effect rather than "deep roots" development. It wasn't that Argentina had fabulous education, efficient and non-corrupt government, robust finances, self-sustaining scientific and technological growth, and so on. It was that they had land, and the stuff land could produce. In the words of Carlos Diaz-Alejandro, they had won the "commodity lottery." If we are beginning our analysis from this point, then of course Argentina seems very rich, and the question is "what the hell happened." But it is a bit of an unusual starting point, in terms of the international economy.
One thing that happened was migration. Land abundance led to high wages, and high wages pulled migrants from Southern and Eastern Europe. This increased the overall size of the Argentine economy, and probably benefited almost everyone, but it suppressed wage growth. (The US also sees this effect, but growth there was so rapid and self-sustaining that it overwhelms the negative effect on wages. Nevertheless, a US without migration would probably have been even richer per person, though much smaller.)
Another thing that happened were international capital flows. There were enormous opportunities to be had in financing the development of railroads, ports, urban infrastructure, meat packing plants, and so on. The stocks and bonds that financed all this were floated in London, and while the Argentine elite took a plenty large share and collaborated closely in these investments, there is no question: the high incomes in Argentina were in many respects a product of the high rates of investment. The money switched on the opportunities, but it also meant that sustaining this level of growth was dependent on continued flows of international capital.
From the 1890s, the fortunes of primary commodity producers starts to shift. Mechanisation comes to agriculture (tractors, harvesters, fertilisers) which creates competition for primary production. Victorian-era "free trade" starts to slide into protectionism. The terms of trade start to shift back towards industrial goods. In some Latin American countries, this led to the beginnings of industrialisation, with imports of bulky and low-value manufactures replaced by local production. But for Argentina, only Buenos Aires was a very large market, and it had both relatively high wages, and a strong trade relationship with Britain, the world's most industrial country. The fall in primary product prices starts to take the edge off of Argentina's economy. Argentina's land advantage was no longer so much of an advantage.
WWI was a big problem for Argentina. Britain had long been the major market for Argentine products, and the source of foreign investment. The war forced Britain to liquidate much of its international portfolio, and was in a far weaker position to make the investments that supported Argentine growth. The US replaced some of that investment, but not all of it. The domestic financial system was fragile, local savings were surprisingly low, the government ran perennial deficits, and growth could only therefore be sustained by injections from abroad.
This might have been fine - developing economies frequently take on large debts - except for the Great Depression. While the depression did not completely demolish the Argentine economy, as it did to Chile, it was a severe shock. Capital flows suddenly stopped and reversed; what was once new and more money flowing in every year, became a burden of repayment every year with no new loans, and difficulty even rolling over the old ones. Tariff barriers went up everywhere, including Britain; this is the context for the Roca-Runcimann treaty, negotiating an Argentine "exception" to the general system of Imperial preference that Britain was establishing in response to the global trade war. (As a side note, Argentina did not entirely default on its debt obligations in the Great Depression, and was largely one of the "well-behaved" creditors of the 1930s. But much like the French reputation for military bravery, popular imagination forgets those bits.)
With the "turning inward" of the global economy in the 1930s, Argentina was on its own to provide internal solutions to the problem of economic growth. Industrialisation, which was already on its way with the increasing size of Buenos Aires and its influx of immigrants, and with the development of "industrial" meat processing, was given a massive boost with the sudden change in the terms of trade. Now industrial products were extremely expensive, and primary products very cheap, pushing investment into manufacturing. Since foreign exchange was at an extreme premium, import substitution was necessity, and eventually policy.
Unfortunately, Argentina was not similarly advantaged as a manufacturer as it had been as an agro-exporter. It was very far from other developed markets, and intra-Latin American trade was anemic. Buenos Aires might have been a lot like Chicago, but then there was no nearby New York, Boston, Philadelphia, or Toronto, and even London seemed more distant with closed world markets. Its educational system and literacy rate were good by Latin American standards, but not so impressive compared with Europe or the other European settler colonies. Its domestic financial system was flimsy and shallow. The political system was... complicated, and the internal question of who should be taxed, and how, was never clearly solved. The advantage that made Argentina rich was in its links with the global economy, and now it had to go it alone.
Nevertheless, Argentina enjoyed a post-WWII boom just like everyone else, with rapidly increasing urbanisation, a transition towards manufacturing as the core of the economy, and an improvement in living standards. This level of growth was sustainable so long as the first stages of industrialisation were being realised. Peron and Peronism represented a populist variant of the sort of urban worker/urban industrialist alliance that characterised many countries in the postwar. Politics was stabilised by spending, and so long as growth continued year on year, the system could keep going. But this was not without cost.
[Cont'd]