Peace of mind means just what it sounds like: a state in which your brain is calm, at ease, and untroubled by worry.
This common phrase describes a feeling of tranquility or state of carefree contentment that can be hard to achieve.
Here are some real-world examples:
He's still a child, emotionally no older than I am; all he wants is happiness and peace of mind. (The Diary of a Young Girl)
He said Context research showed that the most important reason consumers gave for the purchase of smart home products was "to give me peace of mind to know that my home is secure". (BBC)
Piece of mind is incorrect, though it's easy to understand the origin of the error beyond peace and piece being homonyms. There's an expression "to give someone a piece of your mind," which means angrily scolding someone or being bluntly critical so that your unhappiness with whatever they did is abundantly clear. But feeling angry because your neighbor keeps parking in your flower bed is the opposite of peace of mind!
In the examples below, the word piece — a part or section of something — doesn't make any sense. Peace, in the sense of freedom from anxiety or care, is obviously the word that belongs in both sentences.
"Find a place that meets your needs, that provides you flexibility or value or piece of mind." (Washington Times)
"While price is and will always be an issue, it no longer outweighs the personal and financial piece of mind that health insurance coverage provides." (Seattle Times)
You can give someone a piece of your mind when you're irritated and critical, but you'll achieve peace of mind if you let your anger go. Enjoy the peace of mind that comes from knowing the difference between these two phrases!