What could cause skin to bruise with the slightest bump or nick? And then peel away?

Thin skin. Thin skin and increased capillary(very tiny vessels in the skin) fragility can lead to easy bruisability and skin injury from trivial trauma. Are you taking anticoagulants like warfarin or daily aspirin? Make sure you're not lacking in vit. C, vit.K or taking , alleve type medications, corticosteroids, or too much viT.E. The back of the hands and forearms have very thin skin which bruises easily.
It is common. For skin to thin in the sixties. When the skin starts to thin it might look crinkly or wrinkly. If the skin gets bumped or nicked it will get a big bruise easily and sometimes peel . This pattern is usually on the extremities that are easily bumped. It will continue throughout life.