Pending Payment: When your order has been created but payment is not done yet.
Processing: Payment is done and the order is confirmed. This status will remain till the order is delivered. During this time, any tracking info/order updates will be sent via mail/available from the order page in My Account.
Completed: Once the order is shipped to customer, it will be completed. After this, you should be able to see the invoice and download it whenever you want.
Cancelled: The ordered was cancelled or timed out before payment.