What it is
Robert K. Merton coined the term in 1948: “in the beginning, a false definition of the situation evoking a new behavior which makes the originally false conception come true.” His example was a bank run: a rumor that a healthy bank is failing makes everyone pull their money, and then it fails.
The key is that it starts false. The belief is what makes it true.
How it shows up for me
I see it most in how I talk about my kids. If I decide one of them is “the difficult one,” I start treating them that way, and they start living up to it. So I try to be careful with the labels I say out loud.