The only type of skills characters can have within Fire Emblem Echoes are passive or active. Passive skills are usually tied in to the class a character is, but some come with the Equipment they wear. There are three specific types of passive skills called Mounted, Armored or Flier, all of which are listed in a unit's skill list. Active skills typically come with the cost of Health to use, also fall into three separate categories: Combat Arts, Black Magic, and White Magic.
Combat Arts are usually tied into the Equipment a character is wearing. Before they can be used, the unit holding said item must become proficient in its use by simply engaging in combat with the item. Most items have 1-3 Combat Arts tied to them, and each require varying amounts of experience to master them. Usually, the more Combat Arts a item has, the more time it will take to unlock them all. Only the unit that masters the item will be able to use its Combat Arts; a new unit given the same item will have to master it as well before using them. Additionally, Combat Arts can only be used on their corresponding item. If a Combat Art is connected to multiple items, learning it on just one item will allow the unit to master it on other items. Lastly, a character cannot double-attack when a Combat Art is used, though there are some that hit multiple times.
Black Magic is an active skill that can be used by all magic-wielding Classes. Black Magic spells are almost purely offensive, and for the most part, there are only numerical differences between one to next (though a few do carry some special quirks to them). Unlike White Magic and even Combat Arts, all Black Magic spells have a given weight to them. And when casting a Black Magic Spell, the caster's speed will be reduced by said spell's weight.
White Magic, on the contrary, is only usable by Clerics, all promoted magic-wielding Classes (except Harrier), and Celica as a Priestess. While White Magic spells cannot be used to damage enemies, they are instead useful as supporting spells. Typically, White Magic spells are used to heal you unit, but they can also be used to warp units or inflict status damage on opposing foes. Healing spells have a value called Rcv (Recover), which actually indicates how much Health they will recover relative to other healing spells. Finally, some White Magic spells have a range determined by the unit's overall Attack Stat (Atk).
Most Active skills that target enemies will have an associated Attack Power (Mt) and Hit Rate (Hit), some of them may also provide bonuses to the user's Critical Hit Rate, Dodge Rate or Attack Range. For Combat Arts, the Attack Power, Hit Rate, and Critical Rate are added to those of the unit's Standard Attack, while the Avoid Rate will be subtracted from the enemy's Hit Rate. For all types of Magic skills, the same applies to the spell's Attack Power; however, unlike with weapons, the Hit Rate is independent of the caster's Skill Stat. A spell's Accuracy can only be modified by support bonuses and the passive skill of Discipline for Sages.