They are often seen on decaying wood and in soils enriched with wood chips or bark mulch. Cyathus striatus is probably the most commonly encountered species in the temperate northern hemisphere.
The "eggs" are spore cases called peridioles. Peridioles contain glebal tissue, basidia, and basidiospores, and are dispersed by rain. The nests are splash cups. When a raindrop hits one at the right angle the walls are shaped such that the eggs are expelled a good distance from the nest. Each egg has a sticky trailing thread attached to it. If that thread encounters a twig on its flight the egg will swing around and wrap itself around the twig. The spores can then germinate there and start the life cycle over again.