The easiest way to think about how Magento processes all the orders is first knowing that Magento uses an internal order state and a status.
new: This order is new, unprocessed, not picked up by the payment gateway or anything.
pending_payment: Payment is being processed by the payment gateway (this should always automatically resolve if the gateway is set up correctly/is build correctly). Inventory is already reduced.
processing: Invoice has been created, order is ready to be processed
complete: Shipment is made, downloads or giftcard have been mailed.
closed: Order is closed before the invoice has been created. Inventory is returned to stock
canceled: Credit memo has been made.
holded: Manual state so you can hold or unhold the order.
payment_review: Integration with the payment gateway, sometimes a payment has to be reviewed (when a payment is suspected of fraud, PayPal uses this for example).
States in the code: https://github.com/LokeyCoding/magento-mirror/blob/magento-1.8/app/code/core/Mage/Sales/Model/Order.php#L333
Each state in Magento can be mapped to multiple statuses. The statuses is what you can define your self (http://www.magentocommerce.com/knowledge-base/entry/custom-order-status) and you can set a certain status to be the default status.
Payment gateways, connections with other systems usually have the ability to set the status to something different than set by default. Payment gateways sometimes introduce new statuses and use those instead of the default.
In Magento I'd like to have the following procedure: Customer creates an order, I generate an invoice (Status switches to payment_review or something like that), I send out the order (Status switches to completed).
Option 1: Normally when an invoice is created (and no shipment is created) it sets the state to processing and the status to the default status (or some other status if that is set in the payment method).
Option 2: If the shipment is already created or if we are talking about virtual products (gift cards or downloadable products or other virtual products) the state switches to complete and the status to de selected default status (by default also complete).
Currently after creating the invoice the status automatically switches to completed.
It seems we are talking about option 2 here, that means that the order is completely processed. If you wish to do something else with the order after that (maybe you need to do a follow up), you can create a new default status for the complete state and it will automatically switch to that status.
I hope this makes sense.