So you leveled your Unit and you hit the Max Level. What's next? Evolving. The majority, if not all the Units in the game have a higher evolution form, usually much stronger, with a higher level cap.
How To Evolve Units
Once your Unit hits its Max Level (it will say Max, instead of the numbers when Maxed), then you can evolve them. To evolve, go to your Units page, and then tap Evolve (進化).
You should see a layout similar to that of Leveling. You have your Base, Material, and Friend units. Only this time, you are here to evolve. Choose your Max level Unit by selecting it (選択する), and then you should see a different scene.
The Green boxed area in the picture is the Base unit, materials needed and the friend or support chosen to undergo evolution. This is the party you will use to clear the evolution quest.
The Blue boxed area is the friend or support requirement. You can not choose a friend or support that does not meet these requirements. In this scenario the friend/support unit must be a level 50+ Fire unit.
The Yellow boxed area are the coins needed to proceed with the evolving. In this scenario, you would need 51000 coins to proceed. If you do not have the amount of gold, you cannot evolve your Unit.
Once you have the specific unit and friend type chosen, it's now time to run a quest! Yes, to Evolve you need to run a not-too-difficult-at-all quest, and once you are finished with it, you will automatically Evolve.
The dungeon element is decided by the Base Unit, and the only units that will be at your use are the Base Unit, material Unit(s), and Friend Unit. Sadly, this quest does take stamina, but it also gives you exp and gold.
Once you clear the boss, a short cutscene will appear, and it will show the Evolved form of your Base Unit.
Some select ★6 units have the ability to evolve further into even stronger ★7 units. These few units usually received potent power which are almost comparable (but not quite) to the Limited Scratch units.
After reawakening, a unit will gain a new and upgraded set of skills. Some units such as Ryle≒Lancelot even changed their former elemental attribute for another. It is also important to note that even though reawakening gives the units a better Active Skill, the skill level of the previous unit would not transfer to the new skill.
In order to reawaken a unit, special evolution materials called Baetyls are required. These Baetyls are a rare unit drop from the Lostloss Temple quest that is available every JST Sunday. Unlike normal evolution materials there are seven kinds of Baetyls, one for each of the six different elements and a seventh Rainbow Baetyl. This Rainbow Baetyl only appears for a span of one hour right after each of the day's Egg Casino hours are over, in the manner a Guerrilla Boss would appear. Other than the protagonists, all ★7 units need at least one Rainbow Baetyl in order to be reawakened. Also, scratch units require their elemental Baetyls to be evolved into Great Baetyls before being used as materials.
The units available for reawakening seem to be dependent upon their status in the Divine Gate lore. Story-wise deceased units such as the Fairy Queens don't seem to get reawakened forms, while the Protagonists who are said to have received new power in their lores gained one.
List of Units available for reawakening (Updated 1/17/2016)