A Class is, generally, a classification of unit recognized by identical or very similar sprites users of the class share. A class will often specialize in certain stats and/or weapons and will have certain traits that separate them from other classes.
There are many different classes, such as Mages, Pegasus Riders, Wyvern Riders, Lords, Archers, Bards, Dancers, and a vast many others. Each class has different abilities that the player can use to their advantage—Where one class may fail, another may succeed spectacularly. For example, the bard and dancer classes use their abilities to allow a single unit to get an extra turn after they've already moved, and Mages can use their magic on units that physical attacks are ineffective against. Pegasus riders have little limits on where they can move, and while units like Berserkers and Pirates are not quite as mobile, they can still cross water panels (like river and ocean) that can make it difficult for an enemy to escape. Then, although Archers and Snipers cannot fight adjacent units, they can be deadly at long range if the target is kept at a distance. And of course, utilizing knowledge of the Weapon Triangle and Trinity of Magic when deciding on how to use each class can make an enormous difference in battles.
Another facet of classes is the Class Change, where a unit can become a stronger class, either by reaching a certain Level or by using a promotional item (Or both in concert, in many cases). It is normally advised to wait as long as possible before promoting a unit, because once they are promoted, they start over at level one and begin increasing their stats further. Thus, with patience and strategy, a seemingly weak unit can grow into a powerful, useful part of the team.
In Fire Emblem: The Sacred Stones, there is also a system starting with the Trainees, based off the Villager class from Fire Emblem Gaiden. These are units with unique classes that are noticeably weak in comparison with most characters. However, when promoted, they can quickly become a force to be reckoned with. They are Ross' Journeyman class, Ewan's Pupil class, and Amelia's Recruit class. Certain units usually level up in certain stats that their class specializes in, for example: Knights usually specialize in Defense and HP, Fighters in HP and Strength, and Myrmidons in Speed and Skill. Some classes have certain weaknesses like Pegasus Knights who are weak to Bows and Wind Magic. Some characters will have unique statistical gains that separate them from the rest of their class (like Tauroneo who can gain more Resistance than other Generals, and Tanith, who gets more strength than most Falcon Knights).