Horses that are younger than 1 year can be branded.
Inside the stable you can 'Create brand'.
Creating a brand means that you can upload a small picture that will be your brand.
The picture must be 40x40 pixels. It costs you 30000 coins to have a brand (or to replace the one you have).
Once you have purchased the right to brand horses, you will be able to brand the foals you own.
Branding such foals will cost you 15000 coins for each branded foal.
Yes, it is not cheap, but then again, you do not need to brand horses anyway.