Greece is bankrupt. It cannot repay the money it has borrowed from the rest of Europe to finance its deficit spending.
The people of Greece, who enjoyed cheap train fares along with generous government pay and pensions, are now discovering that their government cannot provide what it promised.
Taxes are being raised, government workers are losing their jobs, and they are seeing the retirement age raised beyond 55 and their pensions trimmed. Austerity is not pleasant, and it will last for many years.
Greece also faces higher borrowing costs in the future because its government cannot be trusted to repay its debts. So the burden of financing the debt will be larger for taxpayers in Greece.
Sadly, Portugal, Italy, Ireland, and Spain have similar though not as severe problems, at least as of now (collectively these countries are referred to as the PIIGS). The economy is in shambles, consumer spending is falling, small businesses cannot get loans because the banks have lost too much money lending to the government.
Many banks in Europe lent money to Greece, using depositors' money to buy Greek bonds. Now, most of that money is lost. This means the banks have less money to lend out to businesses and have to tighten up on loans.
This is what happened to large U.S. banks that lent out too much money for bad mortgages. Now, any banks operating in the United States that bought Greek bonds will lose money. Direct U.S. exposure to Greek debt is not very large, but if the governments of other PIIGS countries suffer a similar fate, problems would become more severe.
U.S. debt already exceeds our GDP, although not as much as is the case for the PIIGS. As interest rates rise, the cost of paying interest on the debt becomes a larger and larger burden.