Order statuses are the steps your orders go through. You choose them, so they match how you work: for example New, Processing, Shipped, Delivered. Each change can be e-mailed to the client automatically.
The first status in the list is the one every new order gets, so keep "New" (or your equivalent) at the top. A status that is used by orders cannot be deleted.
An example that suits most shops: New (grey), Paid (blue), Processing (orange), Shipped (purple), Delivered (green), Cancelled (red).
Every status change made by your team sends the Order status updated notification. In Settings → Notifications (category Order) choose who gets it: the client's primary contact, all its contacts, the contact who placed the order, team members or teams. The e-mail includes the order as a PDF and a Show Order button. Its text is in Settings → Email templates → Order status updated.
The automatic move after payment does not send this e-mail.
In their portal under Orders: each order shows its current status, and the order page shows it as a ribbon with the invoices and payments.