These are 'Traits' and bonuses.
The red (T) indicates the trait your horse has.
The bonus numbers are also shown in red like this for example: +13 on skill and 10% on specialization.
The specs trait means that your horse has a 10% bonus for a certain show type.
The skill trait indicates the bonus your horse has on a certain skill.
These bonuses are added to whatever skills/specs your horse has.
On store horses you can set these.
When a new foal is born, it will inherit the way the skills are shared on its parents or the best fit.
The specialization trait of a foal is determined by a 33% chance that it will inherit the 'best spec' trait, 33% chance it will inherit the mare's spec. trait and 33% chance it will inherit the stallion's spec. trait. The 'best spec' would be determined by adding the specialization %'s of the parents and it being the highest % sum.