What is Genetic Potential?

[TofuChild]

And how do you increase it?

[BlackOak2]

And how do you increase it?

Genetic Potential. If you're asking about the number below the Breeder's Report. That is the numerical value (basically) of what the Breeder's Report comments adds up to. Generally speaking, the higher the number, the better your horse is for positive genes. And of course, the lower the value is, the more negative genes your horse has.

For instance, look at these two for comparison:
Twilight Moon

Breeder's Report:This stallion seems as strong as average. Seems to have decent natural speed. Your stallion has a typical amount of stamina. At first glance, I'd say this stallion can't move for grapes. At second glance, I'd second my first glance. This stallion keeps a nice, steady pace! This seems like a very balanced stallion. I've never seen such an agile horse! This stallion is pretty bright.
Movement Intelligence Horse Genetic Potential: 41,290

All About The Blue Crown

Breeder's Report: Doesn't look like he's got much natural strength. I wouldn't get your hopes up on this stallion, speed isn't one of his strengths. I'm surprised this stallion has enough stamina to hold his head up. Cripes, this stallion's feet are all over the place! This stallion's tempo is all over the place! Doesn't seem to be very balanced, trips and stumbles a bit. This stallion doesn't possess much agility... Geez, this stallion is thick as a brick!
Strength Stamina Movement Tempo Intelligence Horse Genetic Potential: 11,832

The first horse offers a number of 41290, while the second offers a number of 11832. Thus the red and green comments can be seen offering insight into the gene potential that the horses carry.

Now how to increase it. Each horse has the ability to increase or decrease this value in all of their offspring. When you cross them with better or worse partners, your number can be heavily influenced to be raised higher, or dropped lower. Also, depending on what those hidden genes contain, a horse may often throw better offspring, or worse offspring, regardless of the partners they're bred with.
The basic rule of thumb, is if you want to raise the number, keep crossing the horse until you find one that has a better breeder's report or a better genetic potential or both, to the parent horse. How you choose to cross them and to what horse, is entirely up to you. And as you can see with the first horse, you can still have red comments on a decently good Breeder's Report and genetic potential (and vice versa).

Also, don't forget that just because a horse has a really high genetic potential, doesn't also mean it's the best at everything. Horses that are considered kind-of mediocre do very well in certain shows and there are a lot of different projects out there that require a lot of different objectives (body types, certain looks, specific colors). It's really easy to find a niche in this game.