When tackling the board-game maps, your marker will move on the tiles or slots based on the number you selected in the roulette. Stopping on an empty tile doesn't do anything. Stopping on an enemy tile triggers a battle. There are also obligatory "STOP" tiles where you have to fight an enemy even when your move can supposedly get past that tile. Intersection tiles are encountered when the branches split to two similar directions. Since you can only move forward, consider taking the path with less threats and more rewards. There are also Trap tiles. These are the tiles with the target cursor on them. Stopping on these tiles will cause random damage to your team. On higher difficulties, the damage is greater. Make sure to avoid these.
Power Up Tiles boosts the Ki of a random character in your party. Though this is not a bad thing, consider this the least of your priority if you can help it. It won't hurt stopping on this tile though if your other available moves lead you to unwanted enemies or traps.
Witch Baba will offer you a random item by paying her a fixed amount indicated on her screen. You have the option to refuse this offer or accept it. Her items are quite uncommon and her price isn't that expensive anyway so there's no hurting in "buying" random items from her.
Item and Zeni tiles are also scattered in the map. Zeni obviously awards you with a random amount of money. The label color of the capsules indicate the type of item. Pink contains Awakening items, Red are Training items, Blue are support/consumables. The color of the capsule itself indicates the rarity of the contents. It's either better quality or quantity. The rarity colors are White (Common), Bronze (Uncommon), Dark Silver (Rare), Gold (Ultra rare).