When you convert an Order into an Invoice in CheckIn, any payments that have been taken on the Order will automatically apply to the Invoice. However, if you need to create a partial Invoice from an Order, the process of applying payment is different.
In an Order for which you have already taken payment, select the item or items you want to Invoice, and click the "Invoice" button in the toolbar.
This will create an Invoice with only the items that have been selected.
Go back to the Order, under the "Deposits" tab. Click on the "Issue Credit" button, which will create a Credit for the open Invoice.
Return to the Invoice, and click on the "Available Credits" button at the top right of the window.
From the "Open Credits" window, select the Credit that was just created by clicking the "Plus" button on the left. (Note that any other Credits on file for this customer will also appear here, as well.)
This will apply the Credit you've selected to the Invoice.
Note that to apply the remainder of the Credit for additional Invoices created from an Order, the same process -- Issue Credit > Select and Apply Credit to Invoice -- also applies.