How can I check whether an online payment actually went through for an order?

Open the order from Orders → Manage Orders and check two things:

• Status: an online order starts as Pending and moves to Processing only when the gateway confirms the payment. If it is still Pending, the payment wasn't confirmed.
• Timeline: a confirmed payment adds a "Payment captured" entry with the amount.

If a customer says they paid but you see neither, look the payment up in your gateway's dashboard before shipping. You can also check Analytics → Reports → Net payments by order.