In Fire Emblem Echoes, a character's class ultimately determines what weapons they can be equipped with. Some classes will also contain passive Skills, which are different movement properties based on being armored or having a mount. When it comes to magical characters, class will only partially determine what spells they can learn: different characters in the same class usually learn a separate set of spells from the next, and they may not acquire the same spells at the exact same level.
Promotion is the process of a unit as a whole ascending to a higher tier class ( or a lower one in a few cases). Promotion usually comes with increased stats or new passive Skills. When a unit does promote, they're again set to Level 1, giving them 19 new potential level ups (Level 20 Cap). Promotion is accomplished a little differently in this game compared to the rest of the Fire Emblem Series, as a character must now be taken to a Mila Statue to promote when their level is high enough to do so.
Villagers have several different promotion options while most other classes only have one, a few of which that are based on the character's gender. Mages will promote to Sage if the character is male and Priestess if the character is female. Dread Fighters, uniquely enough, can promote down to Villager at Level 10. All classes have an ultimate class that is called an overclass, one that they can promote to if you own the DLC corresponding to that given overclass. Each single DLC costs $1.99 and a pack costs $14.99; this DLC is also included in the $44.99 season pass. Found in the free Mila's Bounty 2 DLC is the Pitchfork, an item that can allow any unit outside of Alm and Celica to be promoted to Villager. Only one of these can be obtain, so you need to consider carefully on what unit to use this on.
Each class above Tier 0 will have base stats. If you a unit has any stats below the base stat of the class they are promoted to, those stats will instantly be brought up to the base stat of the given class. Outside of that, the only stat change that comes with a promotion is +1 Health. The only exception to this rule is if a unit is promoted to Villager, which will actually reduce their movement range to 4 if it is higher than such.