The term 'Tome' is an archaic English word literally meaning 'book'. They are appropriately named, as tomes are spell-books from which magic practitioners read to invoke the power of the spells they cast.
In Fire Emblem: Mystery of the Emblem (and by extension Fire Emblem: Ankoku Ryū to Hikari no Tsurugi and Fire Emblem Gaiden) tomes and staves are created by focusing energy into them then using it from the object freely.
In the later games Fire Emblem: Path of Radiance and Fire Emblem: Radiant Dawn it is stated in several instances that tomes are written in 'Old Speech', the original language of the continent these games take place on, Tellius. Old Speech is not mentioned in any other Fire Emblem games, however, a few games state that tomes are written in a lost ancient language. A few characters in Path of Radiance and Radiant Dawn can speak Old Speech, most notably the Heron laguz.
Tomes are used to cast magical attacks. They usually weigh very little to reflect the low strength of magic users. In the GBA Fire Emblem titles there were three types of tomes: Anima, Dark, and Light. These three types of magic form a triangle similar to the Weapon Triangle, known as the Trinity of Magic: Anima beats Light, Light beats Dark, and Dark beats Anima.
In Path of Radiance, Anima magic was further divided into three separate magic classes (Wind, Fire, and Thunder) which form the Anima Triangle: Fire beats Wind, Wind beats Thunder, and Thunder beats Fire. Many new tomes were created for each of these classes. As a result, Dark magic was completely removed so as not to overwhelm the player with too many spells. Light magic is still present in this game, however, unlike in the earlier games, it is neutral against Anima (and vice versa).
In Radiant Dawn, the original magic triangle returned alongside the newer Anima triangle. Dark magic is effective against all three types of Anima magic, which are in turn effective against Light magic.
In Fire Emblem: Awakening, only Anima and Dark magic exists and no light magic exists, except for Book of Naga. No form of the magic triangle exists in the game meaning all tomes deals neutral damage to other tome wielding units.
Tomes appear in the games as small books in the hands of their wielders, but they can be difficult to spot in some of the earlier games due to their lower resolution graphics. Their color varies depending on the spell they cast.