What happens when you owe a bank money will depend on the bank and will also depend on how much money you owe to the bank.
If it's just a few overdraft charges that you owe to the bank and you never pay them then the bank will most likely just close your bank account and then send it to collections to get their money.
Then a collection agency will harass you until you either pay or they might even take you to court.
If you owe a bank a good amount of money as in $500.00 or more then the bank might take you to court and sue you for the debt.
If you owe a bank money and don't pay they'll also report your account to chexsystems which will stay with chexsystems for 5 years and you won't be able to open a new banking account elsewhere for those 5 years unless you get lucky and find a bank that doesn't report or use to chexsystems.
If you're unable to pay back the overdraft charges or the money you owe to the bank you should try and work something out with the bank so your credit rating won't be ruined and cause you more trouble.