These mushrooms are quite small and seem to enjoy hiding in the leaf litter. If it wasn’t for their bright red color I probably would have missed them. The term cup mushroom is a general one describing their chalice or cup shape.
A knowledgeable friend (Thanks, Carla!!!) ID’d this for me. Here is a website:
This photo was taken June 17, 2011, in Green Lane, PA, USA, in a hardwood forest.