There are three types of promotions in the game.
The Kai promotions are usually signalled with a II in the name/card. These are the ones everyone have. Usually it gives a slightly boost to stats and hit points, but it's not to exciting.
Rarer is the Shou promotion. Not everyone has these. Shou promotions upgrade the hime to a new vehicle type. You start over from level one, but the base stats are usually higher as they're using a more powerful vehicle for their armor. Shou promotions can also upgrade again via a Kai promotion.
Rarer still are the Satoru promotions, ie Love Awakening. Unlike the previous two types of promotions, that only require resources, Low Awakening requires the collecting of some rare search drops. Satoru promoted Hime have new armor and gain special abilities that normal Hime dont have. There are no Satoru Kai forms.
Promotion is not done automatically. You have to manually hit the upgrade button when they reach a sufficient level (Which varies from girl to girl.) Currently, as they've changed it in the past, the normal Promotion button is the one 2nd from the left. If it's Orange, you can promote them. Love Awakening is its own button. If they dont have a Satoru form, the button doesnt show up.