These are what all the rankings mean:
Alpha Mare/Stallion: one of each gender is chosen to lead each subherd. They hold all or most of the power in the herd. Their name ends in "spirit".
Deputy Mare/Stallion: one of each gender is chosen by the alpha to "take over" if the alpha retires or dies. They are supportive of the herd and have the power when the alphas are away. Their name ends in "star".
Healers: it doesn't always have to be one of each gender, but it usually is. Healers have a strong connection with the Spritual Herds and have the ability to heal horses with herbs. Their name ends with their herd's name (i.e. in the Moon Herd, a healer's name would end with "moon")
Fighters: from ages 4-10, these horses will fight to protect their herd.
Apprentices: from ages 1-4, apprentices train to become fighters or healers (time period is different for healers). These horses' names end with "hoof".
Broodmares: these mares, from ages 10-20 (or earlier if disabled), provide the herd with new foals. These horse's names end with "light".
Studs: these stallions, from ages 10-20 (or earlier if disabled), breed with the broodmares and provide the herd with new foals. These horse's names end with "dark".
Elders: these retired horses are honored for their support of their herd. They are always protected. They are from ages 20-death. These horses choose their own name.
Foals: these are young horses, taken care by broodmares and/or their mother. Their mother gets to choose the first part of their name, and their name ends with "foal". Once reaching 6 months, their name ends with either "filly" or "colt".