Character potential boards are a system to strengthen characters. The boards are massive and each character has a somewhat unique one.
Each character has a large assortment of new abilities and stat boosts that can be gained by unlocking nodes on a massive character board. You can access the board by clicking the button next to their portrait when in a party.
Each node takes specific items to unlock. Most of them use Vials and Decanters of different colors, which are earned through battles with Descension Bosses. A few require specific items or Mirror Fragments or even Odin Medals to unlock.
The best reward - a permanent increase in number of attacks - requires Yggdrasil Branches or other special items to unlock.
This image shows what a partially-unlocked tree looks like. To unlock an available node, simply select it. The dialog box to the right will show up. Tap the required materials from the bottom frame if you have enough, then click "Yes" to confirm.
Here is the rough outline of each potential board:
There is a character image in the center. Four main diagonal branches (yellow outline) go in each direction. Each branch has a slight lean toward one of the four main stats - Attack, Magic, Defense, or Mind. The further from center the bigger the stat increase from each node. Each branch has locked nodes with unique boosts. There are two such locks per branch. The first one requires 100 Mirror Fragments or a Silver Bangle to unlock it. The second asks for 300 fragments or a Gold Bangle. Attack locks usually increase crit chance and/or crit rate. Magic locks tend to increase damage of a specific element. Defense locks increase guard chances. And Mind locks usually give resist to one of the elements.
Three branches which go from the character avatar straight to the left, up and right (cyan outline) are the ones increasing the number of normal attacks. They start from the big lock which is opened with artifacts/talismans/Yggdrasil Branches. Yggdrasil Branches are available via some events. Aside from Branches, story and collaboration characters can use Talismans which are rarely available as bonuses from gachas. Each weapon type has a different Talisman. Gacha characters can use additional copies of their own artifacts instead. There are 4 additional nodes behind each of these locks giving a large stat boost, as well as a node raising the stat cap.
There is also an elemental damage increase node directly beside the top Attack+ node for each character (purple outline). It can be unlocked with either an Artifact or Talisman depending on the character.
There is also optional branch which goes down from character image (red outline). Not all characters have it. There is only one node there and it gives either Character or Weapon Assist slots for the Party Assist system. Once unlocked it will work for all parties - it doesn't matter if the character in question is in the party.
Below it there is a big blue node for Enchant Skills (green outline). Some characters start with one, others need to learn it first. Here you can let characters learn one, level it up, or change it to something more useful.