What would happen if a country defaulted on its public debt? Could someone explain in full detail to someone that does not understand much what would happen if a country refused to pay its pubic debt?

Someone told me if this happens banks would fall and citizens wouldn't have jobs and savings, but I cannot link these two events. How are these events related? How does one thing lead to another?
A “full explanation” would be extremely long; depending on the situation, a great many things would happen.

For developing countries, defaults are typically on debt in foreign currency, issued under foreign law. They will end up in legal fights with bondholders, and/or end up dealing with the IMF (or other body) to restructure their debt. Economic problems will revolve around being cut off from the financial markets, such as the currency markets.

The scenario you are describing seems to more about countries that issue debt in their own currency, under their own law (for example, the United States, Japan, Canada, etc.) 

For most of these countries, the government is a large borrower. If they refuse to make any payments on debts (which is rare; defaults are often partial), this would represent a loss that is comparable to one year’s production or income (GDP). This would represent severe losses for many savers.

Furthermore, banks hold government bonds to manage their liquidity. Meanwhile, they have very little equity when compared to the size of their balance sheets. Large losses on their bond portfolios would likely result in their bankruptcy. Their bankruptcy would then cause losses on their lenders - bank bond holders, depositors. Furthermore, a bankrupt bank cannot extend new loans, and so any businesses that survived would have a difficult time to borrow to fund expansion. As a result, there would be massive job losses.

In practice, defaults are rarely total (except after wars/revolutions),so we do not see the full potential effects. However, even partial defaults - such as in Greece after the Financial Crisis - typically result is a depression. (I suggest that you look up the Greek debt crisis for an example. The Greek situation is different than developed countries outside the euro, since it does not control the currency it is borrowing in, and so was forced to deal with supranational bodies.)