Action functions in Visualforce allow JavaScript functions to call Apex controller methods using an asynchronous AJAX request. An action function is defined using <apex:actionFunction> and specifies the Apex method to call. The JavaScript function can then invoke the action function to execute the controller method. Action support adds AJAX functionality to a component and calls a controller method based on events, while action functions can be directly called from JavaScript. The example demonstrates using an action function from a checkbox onclick and action support from an input field to call different controller methods and update an output text.