"financial institution," late 15c., originally "money-dealer's counter or shop," from either Old Italian banca or via French banque (itself from the Italian word), both meaning "table," from a Germanic source (such as Old High German bank "bench, moneylender's table"), from Proto-Germanic *bankiz- "shelf," *bankon- (see bank (n.2)). The etymonlogical notion is of the moneylender's exchange table.
As "institution for receiving and lending money" from 1620s. In games of chance, "the sum of money held by the proprietor or one who plays against the rest," by 1720. Bank holiday is from 1871, though the tradition is as old as the Bank of England. To cry all the way to the bank was coined 1956 by U.S. pianist Liberace, after a Madison Square Garden concert that was panned by critics but packed with patrons.
There probably was an Old English cognate but it is not attested in surviving documents. The nasalized form likely is a variant of Old Norse bakki "(river) bank, ridge, mound; cloud bank," cognate with Swedish backe, Danish bakke "hill, rising ground."