What is a database driver exactly that applications use to establish?
You could look at it like this: the API is just the set of commands that you issue to the driver. The driver then maps, or translates those commands into the lower-level commands that it actually understands. The API is often an interface between a specific language and the driver. An ODBC is an API provided by the database vendor to allow interaction with its product. It may provide several versions, one for each language that it wants to support. When you install a printer, one of the things you are doing is installing the API and driver for that printer. The API, for example, knows what commands your work processor wants to send to say, print some red text. The printer may have a different command. The API takes care of this translation.