@achurch It helps to know the German meaning for "Eigen." It means "self" "my own" or refers to an self-view or identity. Depending on context, it is a prefix that can mean "proper" or "unique"
An eigenvector is a unique identifier. It's the ID card of a mathematical system. What's cool about it is that two systems can look very different, but share an eigenvector and eigenvalue. It means that any differences between those two systems are not mathematically important.