Note: Background information originally by Tim Goeke (<firstname.lastname@example.org>)
ODBC (Open Database Connectivity) is an abstract API that allows you to write applications that can interoperate with various RDBMS servers. ODBC provides a product-neutral interface between frontend applications and database servers, allowing a user or developer to write applications that are transportable between servers from different manufacturers..
The ODBC API matches up on the backend to an ODBC-compatible data source. This could be anything from a text file to an Oracle or Postgres RDBMS.
The backend access come from ODBC drivers, or vendor specifc drivers that allow data access. psqlODBC is such a driver, along with others that are available, such as the OpenLink ODBC drivers.
Once you write an ODBC application, you should be able to connect to any back end database, regardless of the vendor, as long as the database schema is the same.
For example. you could have MS SQL Server and Postgres servers that have exactly the same data. Using ODBC, your Windows application would make exactly the same calls and the back end data source would look the same (to the Windows app).