The battlefield quests are both random and straightforward, so there isn't much of a reason to explain them in any sort of detail. For each act you'll receive a new battlefield, accessible via battlefield magical keys. Once you find such a key, you'll be able to enter the battlefield for your act at any time by selecting the battlefield "skill" button under your character portrait.

Each battlefield comes with five shopkeepers who also act as skill trainers. They'll each give you a quest. To solve the quests, all you'll have to do is find the dungeons in the battlefields and make your way to the bottom. The last level of the dungeon is always small and always contains the creature you need to kill or the object you need to find. However, the dungeons start out locked, and you'll only be able to enter the dungeons once you've found a high enough level battlefield magical key.

If you find that the battlefields are boring, you don't have to visit them. Just use them for the shopkeepers. Since you can visit the battlefields at any time, they're perfect for quick trips when you need to sell, identify, or repair. Also, the shopkeepers will change their stock every time you visit the battlefields, so you can move back and forth between the battlefields and main game to hunt for any equipment you might need.

Here are the battlefield maps:

Act I

Act II

Act III

Act IV