There are 4 different DNA alterations right now.
Active state means that the DNA is affecting your horse, passive state means that the parent had the DNA and passed it down BUT it won't have any effect on your horse.
High fertility: When you breed your horse, it has the possibility of foaling twins if the DNA is active (Twins have the same stats but can have different genders and DNA states)
Pegasus: Adds +5 bonus to your horse. This will make your horse jump higher and improve show results (if it is active). This DNA occurs if you add a Pegasus DNA item to your horse or if it is passed down from a parent with peggy DNA.
Linbred: This applies -5 to your horse. This will make your horse jump lower and lower show results, if the DNA is active. This DNA occurs if you breed a horse to one of its parents (The foal will be linebred) or passed down from a parent who had the linebred DNA
Test-tube: This DNA alteration simply means that your horse is a clone. It doesn't have any effect on performance. It just behaves like a clone would before the updates. (Can only be passed down to foals in passive state - doesn't do anything)