It depends on what I am looking for. If I am looking for a broodmare or a performance prospect, they need to have good bloodlines. If they don't have a pedigree, I don't keep looking. They could be the best looking horse I've ever seen, the lack of pedigree will hurt resale or sales of foals. After pedigree, I look at conformation, movement, disposition and such.
If I am looking for a horse for one of my kids to show at lower levels, I look for something that fits the bill. If the pedigree is good, that's a bonus. That type of horse has a different set of criteria for me.