Bloodline is a trait that has a very strong correlation to the racing talent (potential) of a horse. You can find a brief description of this in the Traits section, but this section will discuss it in more detail.
Every horse has one of 5 different bloodline types. The bloodline they have determines how compatible they are when breeding with other horses. This is a very watered-down, simplified way to mimic real life bloodlines (similar to the idea of a 'nick') without the complicated mess that would come from trying to actually make it more realistic. If you've ever heard someone say something like "Chameleon Colors mares tend to cross well with James Bond" (or James Bond sons), that's the sort of thing bloodline type on FFH is all about.
When you breed two horses together, their bloodline type is compared and can either have a positive, negative, or perhaps little to no influence over the foal. Bloodline primarily influences the stats of a horse, raising or lowering them from the 'base' determined by the stat genes. It can also, in bad crosses, result in genetic anomalies in crosses that may have otherwise yielded strongly-gened foals. With that said, you should keep in mind that the producing quality of both parents ultimately matters more than having a perfect bloodline cross. The two work in tandem, so high producers and good bloodline crosses will be the most beneficial.
What makes a cross good or bad? A 'perfect' cross is one in which both parents share a bloodline type. In such a case, the foal may have a bonus to their base stats. When accompanied by high-producing parents, this can result in things like fair/good stat genes that max at 10. In fact, the only way to get a 10 stat is to either have a strong bloodline cross and/or high-producing parents; even Outstanding stat genes cannot alone create a 10. Bloodline types that are only 1 apart will usually have no noticeable influence on the foal. When the types become 2-4 apart, negative impacts may result. At only 2 apart, there may be very minor deductions to stats and a low chance of genetic anomalies. If the types are 4 apart, it will result in significant deductions to stats and a high chance for genetic anomalies.
You can find out a horse's bloodline by buying a Blood Test (in the Pro Shop for PC or the Geneticist for $). You can also figure it out by careful sleuthing through a horse's pedigree (if it has parents). Female Influence determines which parent's bloodline a foal will inherit, assuming both parents do not have the same type. Thus, if your have a horse who has High FI, you know it has its dam's bloodline type. If it has Low FI, you know it has its sire's bloodline type. If it has Average, it could have either. If you trace this back through the pedigree, you can, at the very least, find out (in many cases) which foundation horse it takes after. If you come across a horse in the pedigree that has a Blood Test purchased, then you can know your horse's type for sure.
If you're unsure of type or don't want to spend that much time, you can also try to make the best possible crosses by evaluating family history. If you're breeding a mare, check out her siblings and see what lines they have produced well with. If she has a brother who has produced several stakes foals out of mares going back to a certain stallion, your mare will probably cross well with sons from that same stallion who share his bloodline type. If your mare shares her dam's bloodline type, you can also look at who her dam crossed well with to get an idea of who she might go well with too. The possibilities here are endless, but this is where an attention to detail and intense focus on pedigrees will be beneficial. Always remember - pedigree matters!
You cannot know the bloodline of a cloaked horse and their offspring. A cloak will boost their producer quality so in that case bloodline is not that important. If you can't find the bloodline of your mare it would be a good idea to bred her to a cloaked stud.