Flying Unit is a designation certain classes have that allows them to usually ignore both the bonuses and movement penalties of terrain. Flying units usually receive extra damage from Bows and other types of weapons. Most commonly, Flying Units tend to ride Pegasus and Wyvern thus the Pegasus Knight and Wyvern Rider classes and their related promotions are considered Flying Units.
In all Fire Emblem games except Fire Emblem: Radiant Dawn, all flying units receive bonus damage from bows, ballistae, and Wind-based spells, if present. In Radiant Dawn, all flying units that do not ride Wyverns receive bonus damage from Bows and Wind magic. Wyvern-mounted units receive bonus damage from Thunder magic, not bows or wind magic.
In Radiant Dawn, Crossbows are the ultimate threat to non-wyvern flying units. This is because bonus damage is calculated as triple the might of the weapon, and Crossbows have very high might since they ignore Strength. This means that the entire might of the unit wielding it is tripled rather than just a part of that might.
With regards to terrain, flying units mostly ignore both its benefits and its additional movement cost. There are a few exceptions to this, however, such as in Fire Emblem: The Binding Blade, where flying units expend extra movement to cross Ballistae tiles, or how in most games a flying unit stationed on a Fort will still regain health at the beginning of its turn. Chapter 2-P of Radiant Dawn takes place in a sky arena, with only flying units participating. In this scenario, there are cloud tiles that will inhibit the movement of flying units, as well as providing them with defensive and evasion bonuses.