Adventure mode is a mode unlocked via the Collection tab (indicated by the 3-Cards symbol on the far left of the settings screen. In order to unlock an adventure, one must collect the Monsters needed. For more information about collections, please consult this page or the first link on this page.
Orbs are required to enter an adventure. Orbs can be obtained by filling the Soul Bar. Depending on the length of the mission, the monster required, and the possible difficulty dictate how many orbs are needed to enter. The more orbs a dungeon costs, the better rewards there are a possibility in getting.
Progress Points are acquired by either successfully (or unsuccessfully) completing an adventure. These do not affect gameplay a are merely for scoring purposes (?).
Days are the amount of levels in each adventure. Monsters don't respawn/heal after each day, so make sure you don't accidentally kill one of your monsters. The more days there are, the higher the difficulty. However, the longer the adventure, the more rewards. (Fortune Teller, Daily Sacrifice, etc.)
The "Daily Sacrifice" is an event guaranteed to happen after every successful adventure day. 3 Heroes will appear and you must choose one to be "sacrificed." Depending on which hero you chose to sacrifice, you can get up to 2000 gold (?).
During adventuring, you are limited to the specific monsters used to fill up that collection. Monsters have extremely limited health. Regardless of rarity, monsters are insta-killed by stronger elements, not affected by weaker elements, and take half-damage from their own element. A monster can be healed by "attacking" a castle which has a chance to appear instead of a "Fortune Teller" at the end of a day. To swap monsters, tap the monster that you wish to use before the hero gets halfway across the screen.