A depository institution is any financial institution that accepts deposits from the public. This includes commercial banks, savings banks, credit unions, and savings and loan associations. They take your money, keep it safe, and pay you interest.
What makes depository institutions special? They are the only financial entities that can create money through lending. When a bank accepts a $10,000 deposit and lends out $9,000 of it, that $9,000 gets deposited elsewhere and lent again — this is the fractional reserve system that multiplies money in the economy.
Depository institutions are heavily regulated and typically insured by government agencies — the FDIC in the US (up to $250,000 per depositor) and the DICGC in India (up to Rs 5 lakh). This insurance protects depositors if the institution fails, maintaining public confidence in the banking system.