I am trying to find the name for the following part.
This is underneath a chest that permits it to be moved effortlessly. I am trying to find this part to replace as they are pretty worn.
They are referred to as "Nail on Glides" at my local hardware store. They can be found near the other hardware like the hinges, locks, and door stops.
Not only are they "nail-on" glides, they are "nylon" glides. Similar pronunciation, different meaning. Nylon glides are good for most interior durable hard floors like tile, laminate, linoleum, etc. You'll want to use felt glides for furniture on hardwood (solid or engineered) and other easily-marred floor surfaces (Plexiglas anyone?), and metal glides on concrete, sanded non-skid or other high-hardness abrasive flooring (to avoid eating a softer glide away in a matter of minutes). There are also rubber "glides" (not sure why they call them that) for pretty much any hard surface on which you don't want something to slide around.
Here's the link to a Home Depot search for "furniture glides". What you most likely want, like BMitch said, is these basic cushion glides:
... or maybe these swivel glides for furniture with angled legs: