What does "Free shipping (coupon)" or "Free shipping (group)" mean on an order?

It's the shipping label saved on the order, telling you why shipping was zero:

• Free shipping (coupon): the customer used a coupon that gives free shipping.
• Free shipping (group): the customer belongs to a customer group with free shipping as a perk.

Otherwise the label is your shipping name, for example the Label from Settings → Store Settings → Shipping Setup or the zone method's Label.

You see it next to Shipping in the order's Summary, and on the printed invoice. To check which coupon was used, look for the Coupon line in the same Summary.