There's two places: Moga Village (singleplayer) and Tanzia Port (multiplayer).
Moga Village
-designed for one person, so the monsters don't do a ton of damage and don't have huge amounts of health.
-where the game's "story" is.
-two levels of difficulty:Low and High rank
-one monster exclusive to this mode, as well as certain ranks of monsters exclusive (there's no low rank Arzuros quest in Tanzia, while there is Moga)
-where you obtain your two AI companions
-where your fishing fleet and farm are. While these can be accessed online, can only be upgraded by doing things in the village
-some items can only be obtained here
Tanzia Port
-has quests designed for multiple people so the monsters hit harder and can take more punishment.
-six monsters exclusive to this mode
-three types of the port: Single (by yourself and AI companions), local multiplayer (can only connect to 3ds players and may only bring one AI companion if there's one or two people in the group), and online multiplayer
-three levels of difficulty: Low, High, and G rank
-event and arena quests can only be done here
Completing quests in one mode does not affect the other (with one exception) and any equipment you get in one you can use in the other. You could theoretically do every quest online before you do even one quest offline, and vice versa. When it's all said and done, Moga is like a tutorial for Tanzia, and is also a great way of getting certain items. You really shouldn't ignore either mode.
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