As you probably know, a Lewis acid is a compound that is capable of accepting electron pairs.
If you look at FeBr_3, the first thing that should stand out is the fact that you've got a transition metal, Fe, bonded to a highly electronegative element, Br.
This difference in electronegativity creates a partial positive charge on the Fe, which in turns allows it to accept an electron pair. Remember that transition metals are capable of expanding their octets in order to accomodate more electrons, so a good rule of thumb is that compounds formed by transition metals paired with highly electronegative elements will most likely be Lewis acids.
FeBr_3 is used as a catalyst in the bromination of benzene because of its strong Lewis acid character.
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Without going into detail, the bromine (Br_2) molecule reacts with FeBr_3 and donates a pair of electrons to it. This polarizes Br_2, making it a better electrophile (more attracted to electrons).
