Q: What do they want?
A: When the hedge funds bought the defaulted bonds, they joined the ranks of Argentina's creditors. Now, like lenders everywhere, they want the borrower to repay its debts on the original terms.
The problem is, other creditors had already agreed to cut Argentina a break in 2005 and 2010 by swapping their bonds for new ones worth much less. Investors traded in roughly 93 percent of the defaulted bonds altogether, and this helped Argentina's government slash its debts.
The bonds acquired by Singer's group were among those left over. In 2012, U.S. District Judge Thomas Griesa in New York ordered Argentina to pay the holdouts. They're now owed $1.5 billion in principal and interest.
Q: Why is a US court telling a foreign government what to do?
A: When a big business goes bust, it winds up in bankruptcy court. By contrast, sovereign countries have no dedicated international court to help them strike deals with creditors. So in agreements involving bond sales, language typically stipulates that any legal battle must occur in one of the two biggest financial capitals: New York or London. "That's where the money is," said Anna Gelpern, a professor of international law at Georgetown University and an expert on government debt.
Q: Why does this matter?
A: One worry is that forcing Argentina to pay the holdouts would set a dangerous precedent. The thinking is that it could encourage bondholders to play tough when governments try to restructure their debts. After the Supreme Court's decision, the International Monetary Fund warned that it could have far-reaching repercussions. "We are concerned about possible broader systemic implications," the IMF said.
Another concern is that the ruling upends the usual order of things. In the past, when some creditors had to take precedence over others, sovereign governments typically came before investment funds, said Mark Blyth, a professor of international political economy at Brown University.
"The old hierarchy really no longer applies," he said.
Read MoreArgentine holdouts say would discuss extension
In negotiating with lenders, governments had the threat of default on their side. Creditors accepted restructuring deals in the certainty that they would at least receive something. Now, it seems, "the courts are taking away the possibility of default," Blyth said. "It's part of this wider push to put investors ahead of everyone else."
Q: How big is Argentina's economy?
A: It's South America's second-largest behind Brazil, according to the IMF. The IMF puts the country's economic output for this year at $404 billion, or $9,639 a person. By contrast, the United States has a $17 trillion economy, or $55,000 a person, according to the IMF's data.
Q: Argentina isn't poor. Why doesn't the government just pay off the holdouts and be finished with them?
A: It's not that simple. Argentina is supposed to make an interest payment to bondholders June 30, and the judge's verdict requires it to pay the holdouts their $1.5 billion at the same time. If that were the entire bill, it wouldn't be a problem.
"$1.5 billion isn't going to break Argentina," said Siobhan Morden, head of Latin American strategy at the investment bank Jefferies. "The problem is all the other litigants that could join in."
Paying the hedge funds in full would likely trigger lawsuits from other bondholders demanding to be paid on similar terms. Buenos Aires estimates that the liability could run up to $15 billion. Morden said it could approach $20 billion.
With nearly $29 billion in foreign reserves, Argentina appears to have the money to pay its bills. But those reserves include loans to other countries, deposits with the IMF and other assets that aren't easily used. Take those away, and Argentina has roughly $16 billion on hand.
Struggling countries often find bond investors willing to lend to them to pay other creditors. But Argentina has been locked out of the bond markets for more than a decade. Some investors would probably step up to lend it money - at painfully high interest rates.
"You can see why they have some financial reservations about paying the holdouts," Morden said.