Yes, you can choose to notify a customer when the order status change occurs.
1. This setting is applicable on an order status level, i.e. you can choose which order status changes should trigger email notifications.
To enable notifications, please navigate to Sales -> Order Statuses -> Create a new order status -> E-mail notifications and switch 'Always Notify Customer By E-mail' to 'Yes':
2. Additionally you can control email notifications on an order level. For example, you choose not to send alerts to customers for some particular order statuses changes but you need to notify the customer of a particular order. In this case you can do that from the order view page: