Bank is a financial institution, present in all countries around the world, and an integral part of the world's economy. Banks primarily operate as intermediary institutions, meaning that they consolidate the deposits of one party and turn them into loans for another.
In addition to being intermediary institutions, banks are also money creators. Unlike the government printed money, the Banks issue money mainly in the electronic form. Majority of the money in the United Kingdom is in fact, created by banks. The economists estimate that currently 92% of all world's money is in an electronic form, while only 8% is physical money, or rather, money printed by the government.
When money is borrowed from the bank, the bank creates that money in form of a loan and gives it to the borrower, with a promise to pay back with interest. This is a system by which banks make their profits, and which is why, when the economy isn't doing so well, it may be heard that 'banks aren't lending'. The significance of this is, that when the banks are confident in the economy they will lend as much as possible, creating more money in the process. However, if banks give way too many loans and the debt burden becomes too high, making borrowers unable to pay them back, the banks will stop lending, which in turns limits the money circulating in the economy. As a result, the massive layoffs may happen, the business may close down and the houses may get repossessed to satisfy the large debt, which essentially creates the financial crisis.
On the question of who exactly is Greece is indebted to in 2015, for example, the answer is that much of the debt is actually owed to the banks, such as Deutsche Bank, Alpha Bank, National Bank of Greece, Europbank EFG, European National Central Banks, Eurosystem SMP, Piraeus Bank, Commerzbank, Generali Group and Société Générale, among others. Banks therefore, lend not only to individuals and corporations, but to governments as well.
In the light of the above facts, the banks are indeed 'the good, the bad and the ugly' in the economy. The good in the current system is that the banks pump money into the economy thus moving it forward, the bad is that such system creates an unstable economic model, doomed to periodically crash, and the ugly is that the theory that banks are 'too big to fail' actually hold some truth. If banks circulate money through the economy, then letting them fail is to doom the entire economic system as we know it.