These cups are located at the apex of the leafy moss and function in reproduction. The moss makes little discs of plant tissue inside the cups called gemmae. These gemmae are moved away from the parental plant via a splash-cup dispersal mechanism. It sounds high tech, but really is just using the power of rain. When rain droplets land in the cup the gemmae are dislodged and can be carried in the water as it splatters away from the moss plant. The gemmae may not be dispersed very far, but it is far enough that this structure is advantageous for the plant to have. This is a from of asexual or clonal reproduction. The plant has made a mini copy of itself that can grow into a new moss plant.
The cups are a common and easily recognized feature of the moss genus Tetraphis. There are two species that can be found in North America, Tetraphis geniculata and Tetraphis pellucida. Tetraphis geniculata is rare and grows in limited northern areas on both coasts of North America. It comes as far south as New Hampshire, but has not been sited in Connecticut. The image I have shown is of Tetraphis pellucida (once again taken at the Goodwin State Forest). This species is widespread across temperate areas of North America and is quite common in Connecticut. It grows most commonly on rotten tree parts (logs or branches) on the forest floor. Its common name is the Four-Toothed Moss.