It is very difficult to correctly define a bank, because a bank performs multifarious functions. Different kinds of banks having different functions may be defined in different ways according to their functions. A general and comprehensive definition to cover all types of banking institutions would be unscientific and probably impossible.
Now, Have a look at that some popular definitions of Banking:
According to Keynes:
"A bank is an institute which receives deposits and advances loans."
According to Kinley:
"An establishment which makes to individuals such advances of money or other means of payments as may be required and safely made and to which individuals entrust money or the means of payments when not required for use by them."