The customer order form requires full payment up-front for all new orders, unless you offer flex pay options. However, there are various things an admin can do that result in orders with a nonzero balance due:
To find orders with a balance due, see Payment Due Report.
You can add a payment through the admin interface. On the order detail page, click the dollar plus icon. Fill out the dialogue, and click Add.
Warning: Partial payments can cause accounting discrepancies. When possible it is best to add payments in full.