In Monster Rancher 4 and Monster Rancher EVO, adventures happen automatically as part of the storyline. Once the adventure is over, you cannot return to that place at will like you can with an expedition.
In My Monster Rancher, the adventures your monsters can go on are determined by their Rank. During the adventure, your monster may be able to raise its stats, learn new techniques, or find rare items.
In Monster Farm Online and Monster Farm Lagoon, the quest system works almost the same way, allowing you to choose different adventures that help further the storyline in a sort of "checklist" of what you have completed and not completed. You are usually sent on quests by the town counsel.