ODBC stands for Open Database Connectivity, a standard interface for accessing SQL-based databases. ODBC enables you to implement database independence in an application because any database server that speaks ODBC would respond to the communication from the client in exactly the same way.
You can also use ODBC to connect a database server to a client where there is otherwise no support available. So if MySQL does not include a specific API, you can still use it as your database back end through ODBC.
The ODBC driver for MySQL is called Connector/ODBC and is available as a separate package, downloadable from http://dev.mysql.com/downloads/. The current stable version is 3.51, which implements the ODBC 3.51 specification.