Conformation should be basically correct, if not very correct. Correct being function for the job they are bred to do... While most of the basic conformation points are the same for all breeds, there are breed variances that come into play.
Temperment... Is a huge must for me, mare AND sire. I once owned a little stallion who had a bit of a nasty streak which only seemed to surface with people who were not firm with him... I originally thought it to be human caused, and thought little of it beyond correcting him if he stepped out of line. This was a fabulous little guy, large pony sized, very correct conformation, nice lines.... And an apparently hereditary mean streak. He passed the trait to four out of five offspring, I own the fifth (large pony gelding with a fantastic personality)... Later, after I had already gelded and sold him, I found out his sire was known for his bad attitude and hard to manage temperment, which sometimes seemed to show up in his offspring. So some temperment can be passed from the sire, in some cases!
Ability... I don't really care if a horse has never been shown, personally I think there is more to life than showing, but the horse does need to have a job it is GOOD at, or better yet, excels at.
Pedigree... I would never breed on pedigree alone, but it is never a bad idea to look over previous generations for genetic traits which may be desireable, or not.
For breeds where it is an issue... DNA testing for hereditary diseases is a huge must. Now that we know these things are definitely genetic, there is no excuse not to test and refrain from breeding horses who are known carriers of such. (HYPP, HERDA, PSSM (I think is the abbreviation) ...to name a few)
There is probably stuff I am not thinking of right now... But those hit the top of my list.