Publishing an invoice is how you transfer or post an invoice to your accounting package.
Within the Inbox, select the Supplier account, Category and Tax codes. Click on the green 'tick' to publish the invoice to your accounts package. The icon turns solid green to indicate it is being published.
Once published in your accounts package an icon appears below the publish button to indicate that the transfer is complete. Published to Xero in this example:
From within the single invoice view click on Publish and Next. This will publish the current invoice being viewed and bring up the next invoice in your inbox.