Brave abilities are abilities which can be awakened to become even stronger. Brave abilities are displayed with a different color icon on which their awakening level is displayed.
To awaken your abilities, go to "Units" menu from home screen, and then tap the "Awakenable/Latent Abilities" button. Units with awakenable abilities can be selected. Alternatively if the unit you want to ability awaken is in a party, select that unit, tap the "NV Menu" button, and then tap the "Brave Ability Enhancement" button. Brave ability awakening materials, Brave Insignias and Gil are required to awaken an ability.
There might also be other events, bundle, and promotion to give out these materials, but those 3 are the most regular way to obtain them.
Brave insignias are a material required to awaken abilities from Lv.3-5. They are normally obtained from the Challenge of the Brave or Chronicle Battle events found in the Challenges tab of the Vortex. There may be other events and promotions to give out these materials, but Challenges are the primary way to obtain them. Insignias are unit-specific unless otherwise noted.