The elections in Greece are being described as a rejection of the austerity measures imposed by "the Troika" — the European Central Bank, the EU, and the International Monetary Fund, which required steep budget cuts as a condition for the Greek bailout.
But if Greece were to abandon the Troika, it could face more austerity, not less.
The government of Greece spends more money than it raises in taxes. It can do this because it borrows money. But the only people willing and able to lend money to Greece are the people running Troika.
If the new government in Greece rejected the Troika, there would be no one left for Greece to borrow from. Greece would not be able to keep paying its debts, and might choose (or be forced) to leave the euro.
This would lead to a collapse of the Greek banking system, which holds lots of Greek government debt. It would lead to a period of chaos for Greek companies that do business abroad. The economy, which is already awful, would get even worse.
Some people think this is actually Greece's least-bad option in the long run. Leaving the euro would allow Greece to devalue its currency, making its exports more competitive. Defaulting on all its debt would allow the government to stop spending so much on interest payments.
Maybe. But in the short run, it would be an economic disaster for Greece. And it would require the government to cut spending even more, as the Greek economy continued to collapse, and foreign lenders abandoned the country.