Every paid order shows up in your conversation with Logan, and you can also see them by asking him directly.
Order statuses
Paid — the customer has paid but you have not shipped yet
Fulfilled — you have shipped the order
Refunded — money has been returned to the customer
Canceled — the order was canceled before fulfillment
Fulfill an order
Tell Logan: "Mark order 1032 as fulfilled with tracking 1Z999AA10123456784". You can also open the order, confirm the line items, pack the shipment and click Mark as fulfilled. Minicart emails the customer a shipping confirmation automatically.
Partial fulfillment
If only some line items are shipping, fulfill just those. The rest of the order stays open until you ship them.
Logan keeps you on pace
Logan flags orders that have been sitting unfulfilled longer than your usual turnaround. Ask him "what still needs to ship?" for a quick summary.