An offshore bank is a bank located outside the country of residence of the depositor, typically in a low tax jurisdiction (or tax haven) that provides financial and legal advantages. These advantages typically include:
greater privacy (see also bank secrecy, a principle born with the 1934 Swiss Banking Act)
low or no taxation (i.e. tax havens)
easy access to deposits (at least in terms of regulation)
protection against local political or financial instability
While the term originates from the Channel Islands being "offshore" from the United Kingdom, and most offshore banks are located in island nations to this day, the term is used figuratively to refer to such banks regardless of location, including Swiss banks and those of other landlocked nations such as Luxembourg and Andorra.