How does it play out when you and your significant other are forced to agree to disagree?

Being two different people of course the individuals in a couple are not always going to agree on all of the big stuff. So when you and your spouse or S/O hit one of those issues and are forced to agree to disagree, what does it look like? How do you get past the potential resentment for not having the other person on your side? Does it faze you that they might not approve of your feelings on the subject?