Character classes play a great role in the game as this gives characters access to powerful skills as well as sets their level caps, which in turn affects their growth and stats. Most characters you acquire early on start with "base" classes. Base classes can unlock two skills, one at LV1 (default) and LV10 while Advanced Classes will unlock their skills at LV5 and LV15. There are Special Classes (Songstress, Maid, Butler) that doesn't have advanced versions where the level cap is 40. Skills are learned in similar intervals as well.
Once the level cap is reached, the unit will stop earning EXP and since it can't level up anymore, no further skills can be learned even if you change classes. (Only base to advance class promotion resets the stats and level) There is a seal however, called an Eternal Seal that will increase the level cap of a maxed-level unit by 5. This is ideal if you want that unit to re-class and learn more skills.
At LV10, you can use a Master Seal to change them to their advanced classes. However, it is recommended to refrain from doing so and wait until you reach the maximum level for the base class (LV20). Changing to an advanced class will reset the level back to LV1 but will retain the stats the unit has acquired so far. A LV20 base class will obviously have higher stats than a LV10 base class turned into an advanced class.
Depending in your progress in the story (around Chapter 18 onwards), offspring characters you'll get will have an Offspring Seal in their inventory. These special seals will allow the offspring to change to an advanced class. They'll start with their parent's base class but at max level. Offspring seals functions the same as a Master Seal but only offspring characters can use them.
Each base class can transform into two branching advanced classes. For example, a Norhian Cavalier can advanced to either a Paladin or Great Knight. A Hoshidan Sky Knight can transform to a Falcon Knight or Kinshi Knight. A Dark Mage can become a Dark Knight or Sorcerer. Hoshido and Nohr will have unique but fairly similar classes. The avatar will always start off as a Nohrian Prince/Princess but can be later changed to a Hoshidan or Nohrian Noble. Since the avatar can pretty much change into a wide array of classes, it's possible for him/her to learn a very diverse skill set.
For more information about all the classes in the game, please refer to the dedicated section in this guide.