Inherit is a place where you can transfer some different power sections of one of your Pokémon to another one.
Monster Riot is a daily game mode that starts every day at 21:00h. This is one of the main sources of Diamonds for the players.
World Boss is a daily game mode that starts every day at 20:00h, and where players compete to output the biggest damage possible while killing a Boss.
Crystal Collection is a daily game mode where you try do as most damage as possible to a Crystal to get Gold in return.
Pokémon League, also previously known as Monster League is a daily game mode where you have 15 islands to defeat, each island having a real player but with modified power.
Home is unlocked at lvl29. It's one of the most important features of the game when you're low level.