In a broad sense, people refer to offshore banking as the services that an individual receives from a bank outside his or her country of residence. For instance, if I live in the USA and have a deposit in a bank in the UK, it can be said that I am an offshore banking user.
In the narrower sense by offshore banking people mean services that an individual receives in an offshore jurisdiction also referred as a tax haven. Offshore jurisdictions are countries with low tax structures and their legislation ensures bank secrecy and privacy. There are number of offshore jurisdictions in the world like Madeira, Monaco, Belize, Guernsey, Panama, the Bahamas, Antigua and Barbuda, Singapore, Bahrain, Seychelles, Isle of Man, Luxembourg, Malta, Jersey, Switzerland, the Cayman Islands, Ireland, Gibraltar, Andorra, Hong Kong, Barbados, Labuan, Anguilla, and Bermuda.
Offshore banking is often associated with terrorism-financing, criminal activities and tax evasion, but this is the common mistake. It is a legal service provided by licensed financial institution in offshore jurisdictions. What can be illegal are the actions of an individual using advantages of such services for illegal and criminal affairs. For instance, if you have an offshore bank account and you receive an income on it, you have to report it to your country's tax department if your country's legislation requires so. Otherwise you will avoid taxes and that will be classified as tax evasion. In other words, having offshore bank account is not illegal, but hiding taxes using such an account is illegal and punishable.
Offshore bank accounts have several advantages. First of all it's bank secrecy - no one can access your personal information and your offshore bank has no right to provide information to third parties, unless there is a decision of a local court. Usually such decisions are extremely difficult to acquire, and that makes your information unreachable.