Field Mode
Field Mode is the new method of exploration in Neopets Ultima. It includes you and your party, the area your exploring, and of course your map. In Field Mode, you can access the Neopian Database (ND), which contains the special features that were popular in the original Neopets, like message boards, the news page, the Neopedia, and let's not forget the ol' Neopian Times. In Field Mode, you can shop, interact with other users, and explore Neopia. Plus, you can also run into fights (usually at random) when in Field Mode. And you can form parties with your active pets, as well as other users and their pets. The fields also have things like Neohomes (which now accually serve a purpose).
Battle Mode
When you run into a fight the gameplay automatically shifts into Battle Mode. Unlike the Battledome style used in the original Neopets, Battle Mode is pretty advanced. The design is kind of a blend of the battle systems of Chrono Cross and Final Fantasy X, and since you'll travel in a party (hopefully), you'll now have allies to help you in battle. On your turn, you can choose to attack, use an ability, use an item, or guard against attacks. Attacking with certain weapons, or using certain abilities or items can cause status effects on the target.