When our important rights clash with one another, we can either seek some form of compromise or simply deny one of the rights. This often leads to discord within our country.
One example of this is the fight over gay rights today. Advocates of things like gay marriage argue that all people in a democracy should be treated equally. This is a fundamental right in a democracy. However, the right to majority rule is another fundamental right in a democracy. Therefore, some people feel they should be allowed to vote to ban gay marriage. In this way, two rights (the right to be treated equally and the right to majority rule) come into conflict. We have not yet decided as a society how we will resolve this conflict.