Performance Ratings can be purchased in a bulk stablewide option at the Pro Shop. These will show up for any horses you currently own, as well as any horses you own in the future. They will appear in race notes when a horse races, both on their race history page and in the race itself. They will appear as [PR: ##].
A Performance Rating tells you exactly how your horse did in a race. The higher the number, the better your horse ran. The PR adds in several different factors (explained in the Racing section of the handbook), so it will usually look fairly different from the potential. It adds in the 6 main stats, but also factors in the two doubled stats for a racing style. The most a horse can earn from their 6 main stats is 80, counting in two doubled stats. It also adds in factors like courage, filly power, colt composure, distance bonuses, jockey familiarity, home track bonuses, consistency, health, maturity, and more. Any score over 100 is considered very good, typically placing a horse in G3+ races!
If your horse is earning low scores, you might want to check it's race notes and make sure there are no problems there. If there aren't, you may have a horse who has low drive (little desire to race). If everything else checks out good, your horse may just have low potential. You could consider gelding the horse, or just drop it into lower classes of races where it can actually be competitive. A filly who is running poorly against colts may show noticeable improvements when running in filly only races!