Fertility: This gene determines how easily a horse gets in foal, and how likely a mare is to carry a foal to term. Both parents' fertility factors in when breeding! If a horse has low fertility, you may opt to feed one parent Vigor to guarantee the mare will get in foal.
Heritage: This is a hidden quad-gene that loosely influences the ability of a horse to pass on genes to their offspring. A horse with high heritage is more likely to pass on positive genes (X's), while a horse with bad heritage is more likely to pass on bad genes (x's). This makes it possible for a horse with low(ish) genes to actually produce high-quality foals. Each parent's' Heritage is evaluated - NOT the foal's. Heritage will usually not allow a horse to pass anything that they are not genetically capable of - a horse with XX cannot throw an x and a horse with xx cannot throw an X (with the exception of bad Heritage, poor bloodline cross, or high pregnancy risk, both of which can introduce x genes where the parents had none).
Male Probability: This hidden gene determines the likelihood of having male offspring. The gender of a foal is determined by the MP gene of that foal, not its parents. A foal born with MP is more likely to be a colt, and a foal born with mp is more likely to be a filly. This is not 100% guaranteed, so it's possible to have males with mp or females with MP.
Female Influence: This hidden gene determines how influential a female's bloodline is on her offspring. If this is homozygous in the foal, the foal will usually follow their dam's bloodline. If it is recessive in the foal, they will usually follow their sire's bloodline.
Producer Quality: This hidden gene determines a horse's likelihood of producing high potential foals. You cannot see producer quality, but only infer it based on the results. Producer quality applies to both mares and studs, but carries a far greater weight for studs.