Offshore banking as the name suggests is a financial institution which is not within the shores of a country of depositor of funds rather it is outside the shores. The question is why people go for this type of banking well the answer for it is simple, due to tax related benefit.
This service is used by people who have money on which they have not paid taxes to their government and in order to save tax they resort to depositing the money to banks which are situated outside the country and are considered to be tax havens. This is the reason why many nations are trying to bring back this money back to their countries as money deposited in these banks is basically tax evasion which is a big loss for the government and people of any country. In simple words offshore banking is for big industrialist and politicians who have unaccounted money and not for normal people like me and you pay taxes regularly and promptly.