Briefly stated, what I'm calling the form refresh is what happens when Drupal finally asks the callback function pictured above what response to send to the client. This can be one of two things:
- A piece of the form that will be rendered to replace the contents of the wrapper element specified in the #ajax array
- An array of AJAX commands to execute in the browser
We do both in Drupal Commerce, sometimes simply refreshing part of the form that has been updated and other times sending back a nice fat batch of commands to update various parts of the page. We use commands arrays to great effect to make attribute widgets on the Add to Cart form dynamically update product fields that have been rendered into node pages. This is how product images change based on the color selected, for example.
To create these commands, you use functions like ajax_command_replace(), which simply swaps out the contents of an element on the page with something else as if you'd used the first type of response. The other commands are mostly similar manipulations of the DOM, with the full list of commands available here.
I personally think we need more modules brave enough to use ajax_command_alert() in their #ajax callbacks.
This diff from the Commerce Shipping module shows this (hack?) in action. Theoretically, ajax_command_invoke() invokes a jQuery method on some element selected from the DOM, but this example currently just triggers another element's click event using a small jQuery plugin that breaks a design rule by returning nothing to chain to. C'est la vie.