What are the parts of the tundra ecosystems?

The details will depend on which tundra ecosystem one is referring to, but tundra are characterized by cold temperatures, little precipitation, and no trees.
Tundras are found in the Arctic and above the treeline on very high mountains. There are three general categories of tundras: alpine, Arctic, and Antarctic.
Areas classified as tundra are purple in the image below.
Tundras are characterized by cold temperatures, little precipitation, a short growing season, and no trees. Tundras also experience freeze and thaw cycles, as opposed to polar regions which are always frozen.
Plants found in this ecosystem include shrubs, mosses, grasses, and lichen. Animals include mountain goats, pikas, voles, caribou, arctic foxes, elk, and others.
Arctic tundra is characterized by permafrost, permanently frozen subsoil. In contrast, alpine tundra does not have permafrost and has well-drained soil. The tundra in Antarctica has more available water and thus slightly higher primary production than the Arctic tundra but no terrestrial mammals.
To read extensively about the tundra, see this webpage by Encyclopedia Britannica.