From my understanding, a metric defines a more abstract entity than a norm, but I don't feel like I truly understand. Can someone please explain it to me in layman's terms?
A norm is a concept that only makes sense when you have a vector space. It defines the notion of the magnitude of vectors and can be used to measure the distance between two vectors as the magnitude of its difference. Norms are linear in that they preserve (positive) scaling. This means that if you scale (zoom) down or up a configuration of vectors (an operation that only makes sense in a vector space), the distances between the vectors will be scaled in the same proportion.
A metric is a more general notion that can be predicated on spaces where there is no underlying algebraic structure. They incarnate the concept of distance with independence from any algebraic features (which might not even exist in these spaces). If you have a norm, you have a distance, but you can have a distance without having any sum operation or scalar action.
There is a third level of abstraction where the concept of proximity can be expressed without any distance. These are called topological spaces and their embodiment does not rely on the concept of distance (or norms) but on the concept of neighborhood.