In the real world, differences in drivers and the level of ODBC support lessens the potential of ODBC:
Access, Delphi, and Visual Basic all support ODBC directly.
Under C++, such as Visual C++, you can use the C++ ODBC API.
In Visual C++, you can use the CRecordSet class, which wraps the ODBC API set within an MFC 4.2 class. This is the easiest route if you are doing Windows C++ development under Windows NT.
" If I write an application for Postgres can I write it using ODBC calls to the Postgres server, or is that only when another database program like MS SQL Server or Access needs to access the data? "
The ODBC API is the way to go. For Visual C++ coding you can find out more at Microsoft's web site or in your VC++ docs.
Visual Basic and the other RAD tools have Recordset objects that use ODBC directly to access data. Using the data-aware controls, you can quickly link to the ODBC back end database (very quickly).
Playing around with MS Access will help you sort this out. Try using File->Get External Data.
Tip: You'll have to set up a DSN first.