Bloodlines for your discipline are helpful. You wouldn't use a Hanoverian for a champion reining horse and you wouldn't ask quarter horse to do a GP dressage test.
Bloodlines give you a good set up for success and let you understand what you are working with ahead of time to a small extent. Some lines are known for having difficult horses, others are known for having horses that are mellow and easy to work with.
The best breedings are between two well-built horses who compliment eachother and have been successful in their own right. It doesn't mean they will throw all champions or another horse couldn't be a champion, but chances are higher when you are selectively breeding to get what you want (ie: a champion dressage horse. or whatever).
When it comes to showing, registry is an issue for certain things. With AQHA if you want to be the next western pleasure champion, your horse is going to need to be a quarter horse. Racehorses need to be registered to the jockey club (or AQHA or Arab registry), etc.
But excluding those factors, what is MOST important is heart, and confirmation.
If you have a horse that is built for the sport and has the drive and heart to do it, you're at a better place than a horse with a good pedigree who hates his job.