Is the Drush backend API a web-accessible API? How does one install it?
As the previous respondents said, drush is a cli tool.  If you would like to invoke Drupal functions via web services, try the services module.  If you would like to manage multiple Drupal sites via a web ui, try aegir.

If you would like to call drush functions remotely from php:

$values = drush_invoke_process('@alias', 'drush-command', $args_array, $options_array);


See examples/example.aliases.drushrc.php for examples on how to make an alias record.  Instead of '@alias', you can also pass in an array containing the site record definition, or you can use username@myserver.com/path/to/drupal#mysite.com.  See the source code comments in drush for an explanation of drush_invoke_process, and the contents of the $values array.