Every race comes with a list of armors that can be equipped, skills that are enabled for use, and abilities for passive benefits. Races also give additional skills and abilities by leveling them up, through Job EXP via battling or consuming seeds. A race is considered "mastered" at level 10 and cannot level further, but most races unlock more advanced versions once mastered. Mastered races are marked by a star at the race selection screen.
For most characters, race is the key factor that defines which armor said character can wear. The only exception are humans, which are a case by case basis.
Races can be freely changed by the priest at the Ilias Temple.
Some races require a Race Change Item before they can be selected which are found throughout various locations in the world.
Basic Races are the weakest form available to characters.
Intermediate Races are when you start choosing what sort of role you want your character to play.
Advanced Races are when things really heat up.
Sealed Races are the strongest forms your companions will take. They are not yet available, and will be unlocked in Part 3.