Just wondering what this means.

If you are meaning by the horse names in the the barns, R means it needs a rider a heart means it's up for breeding, E means in in an event, and A means it's up for auction. If you mean by the horses names in the stable page where you can change the name, it means that the names might've been in use and they want that name for the horse to make it unique to the game code. It could also be a mark for a specific stable. I'm bad at explaining but, hope this helps!

from the game guide:

• - Your horse needs to be assigned to a rider.
• - Your horse has been entered into a regular show.
• - Your horse has been entered into a junior rider eventing rally.
• - Your horse is up for breeding.
• - Your horse is for sale.
• - Your horse is in the auction.
• - Your mare is on Cycle Suppressant.
• - Your horse has enough Federation points to enter HEWG Qualifier Shows.
• - Your yearling is entered into a Futurity.
• - Your horse is a member of the JRA.
• - Your horse has been entered into a HEWG Qualifier Show.
• - Your horse has been entered into the dressage musical freestyle.
• - Your horse has been entered into the indoor cross country competition.
• - Your horse has been entered into the puissance wall competition.
• - Your horse's show schedule is being managed by the Barn Manager.