In Monster Rancher 2, your monster travels along a set path that has 3 or 4 parts to it and you play as the monster, completing tasks along the way. At the end of the errantry there is usually a boss monster to defeat.
In the DS games, errantries are called Drills (which is a bit confusing, but oh well!) and your monster can travel across spaces on a map by rolling dice to see how far it can go, much like the spaces on a board game. The map changes depending on which of the four seasons you drill your monster during.