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In  a Barter system there is no use of money in trade.  Two traders or people exchange goods or services that they want from each other.  Each one agrees to the goods or the services that the other is offering.  So this is the situation of coincidence of their wants being satisfied by the other.

This situation of each other agreeing to buy and sell their goods or services is called double coincidence of Wants.

In Barter system of trade, Double coincidence of wants is a must for exchange to happen.

In a bank, when I deposit money, I agree to what they offer me (their certificate, the interest and safety of my deposit and their current services at the same or other branches or ATMs).  They agree to what I offer (my money, and my promise to hold deposit).

So, this is a double coincidence of wants.

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In simple words, co incidence of wants, is meeting a person who wants your item or thing and that same person also desires what you have. Thus, a successful and satisfactory exchange of things occurs!