Sometimes in battle a large circle appears in the middle of the screen saying press B. It is obviously some timer but I can't figure out what prompts it. I figured it might be related to crits, but it doesn't always correlate with anybody using an ability. What is its purpose?
The circle is called Soul Challenge. It appears when the character you are playing(1) initiates Soul Voice. Matching the white ring yields a perfect pass and matching the inner blue circle yields a good pass. Earlier or later than these yields a failure. When you pass the Soul Challenge, your squad can respond by using an appropriate response art. As with all successful Soul Voice responses, this will heal both participants. If one of the participants (initiator or responder) is the main character, affinity will increase.
Passing the Soul Challenge raises Morale (the three small white bars just above the left of the arts bar). As Morale improves, your team has a higher chance to initiate Soul Voice for you to respond to.
All characters have 20 triggers which have a chance to activate Soul Voice. These are found in the Soul Voice menu. For your character, the first 16 triggers can only be customized in terms of the response and effect, while the last 4 triggers can also have customized activation conditions. For the other characters, the triggers are slightly less customizable - Elma would like you to respond with melee.
One of the custom triggers is to get 3 crits in a row.
(1) You can play other characters by putting them in the first party slot. The played character will activate Soul Challenges.
You know when someone mentions for you to use a certain art? Such as "Move in close range!" these are called soul voices, when these are mentioned a certain colour of art in your bar will get an aura around it and when that art is used you will get a healing bonus to the team and also a buff on the art of some sort. What the circles with press B are is you using your soul voice for the rest of the team to use a art and heal back from It and get a buff, perfect grants TP also.
When you are prompted to press B and your timing is right:
If you get it in the blue, you get "Good". If you get it in the white, you get "Perfect". Perfect gives a TP bonus.
Completing these combos and soul challenges grant bonus effects, damage, and they always heal. Source
This also hints at the healing since they mention that people do not press B on purpose to keep their health low. This can trigger some extra effects with some skills/arts.