I found a new bow (Rank B) on the field, and while I've given it to my Archer (Rank C), he can't seem to use it during a fight. What determines whether a unit can use an item?
The weapon has a ranking on it and the character has a weapon ranking. If the character weapon ranking is not greater to or equal to the weapon's then they cannot use it. The ranking goes from E (lowest) to A (highest).
In the bottom left hand corner you can see the characters weapon rank.
No luck finding a picture of a weapon description though.
Additionally to @Tony's answer, it is good to note, that there are some weapons, that can't be used by just anybody, even if that character meets the weapon level requirement.
Those weapons are:
As well as any Dark Tome, which can only be used by Dark Mages, Sorcerers and any other tome user with the skill Shadowgift equipped, that is: Aversa and if married to the Avatar, Morgan as well. Some enemies and DLC characters may possess the Shadowgift skill as well, such as Micaiah and Katarina.