Short answers, pulled from the story.
A plain is a flat expanse of land with few rapid or steep changes in elevation. Plains occur as lowlands along valleys or at the base of mountains, as coastal plains, and as plateaus or uplands. They are present on all continents and cover more than one-third of the world's land area.
Plains form through several processes: deposition of sediment by water (alluvial and flood plains), movement of glaciers (outwash and till plains), accumulation of lava flows (lava plains), erosion of hills and mountains (peneplains and pediplains), and geological subsidence (structural plains). Lacustrine plains originally formed as lake beds.
Biomes on plains include grassland (temperate or subtropical), steppe (semi-arid), savannah (tropical), and tundra (polar). In some cases deserts are also considered plains.
Plains formed from water-deposited sediments tend to have deep, fertile soils. Their flatness also makes mechanization of crop production easier. Grassland plains additionally support livestock grazing.
A flood plain is land adjacent to a river, lake, or wetland that experiences flooding fairly regularly in the present or recent past. An alluvial plain is a broader category that includes current flood plains as well as areas where flooding occurred historically, including zones that may flood only a few times per century.
Planitia is the Latin word for plain and is used to name flat regions on extraterrestrial bodies such as planets and moons. Examples include Hellas Planitia on Mars and Sedna Planitia on Venus.