I have an order that was processed through paypal. In the admin panel, the note says: Your customer has cancelled their payment with PayPal. An Email was already sent, thanking them for their order,
at the time of checkout. You will need to update this order status to "Cancelled" in order for
them to receive a cancellation email and update your inventory.

When I went to paypal, the money was processed to my account and does not appear to have been canceled. How do I know which to believe?

I'd contact the customer to find out what they did then contact PayPal to see if there are any pending cancellations or credits to your account. If you still aren't sure, wait a few more days then ship the order.