Since political ideas only matter when embodied by living persons, it stands to reason that the day-to-day battle between these ideas comes down to sheer numbers of supporters. Two primary methods are used to create new supporters. One can simply convert others to your side, through rhetoric or force. Rhetoric is the most noble, as it allows people to make up their own minds, and presumably some percentage of those people will see through cheap rhetoric to the substance of the position, and thus better ideas will survive and weaker ones will die. However, no communication is perfect, and the rhetoric itself can sell a position, as DemaGogues know.
Force is more common than many would like to believe. Converting the conquered to your religion by the sword (or gun barrel) is the first image that comes to mind because it never seems to go out of fashion. Religions are less interesting, though, than economic systems of trade, and the 20th century was marked often by wars over what amounts to banking systems (e.g. the Cold War). Of course, economic warfare is slightly nicer than religious warfare, because blood doesn't need to be spilled today as long as food can be withheld until starvation hits tomorrow. The dominant powers of the world always wait for the 'barbarians' to stop being so 'barbaric' and come on side as a client state in the glorious economy of the master state.
The second primary method of creating new supporters is to simply create new people. Children are the future, and by far one of the most fought over pawns in the political world (as any parent knows from their local school board meeting). Since children are easily imprintable, and they will grow up to one day become at least passive supporters if not active supporters or even fanatical evangelical supporters of your idea, it becomes imperative for the leaders of any political idea to implant their thought onto the new-born generation. It is no surprise that most children are the religion of their parents. You are a Christian, Muslim, Jew, Mormon, Hindu, Buddhist, Jain, or whatever by mere accident of birth. Some religions actually accept this as fact and incorporate it into their ontology of human affairs (e.g. Hinduism, Jainism), but most simply demand that the rest of the world conform to their religion. Since a religion is usually bound to some economic trading system, there is a good reason for this to happen.
If you cannot create new supporters, you can always destroy supporters of competing viewpoints, which is typically called war. This fact is not hidden; in fact, it is trumpeted as the motivation of going to war, as our society must destroy the evil dark ideology of our enemies that threatens to destroy the future of our children (who are embodiments of our ideals and values, if you recall). The Allies had to destroy fascism. The West had to destroy communism. After September 11, 2001, the West is trying to destroy 'Islamic extremists' or whatever term they invent this week until the demonized term has enough rhetorical stickiness.
Ignoble as this may seem, if two political ideas are incompatible, and one side is both unpersuadable and hostile, then one can only destroy the people who believe in those ideas. The trouble is that there are often a lot of people who believe in those ideas to some degree, and the more that idea is threatened, the more they are likely to defend the idea as an expression of their freedom to hold that idea. Therefore, you cannot merely kill the leaders of a political idea, but you must destroy the entire militarized society that supported the political idea. Since this outcome is so gruesome, this avenue must be truly necessary rather than flippantly chosen — for without the stomach to carry the fight to the very end of your enemy's society's destruction, the result will only be a hardening of your opponents' will to defeat you, and they will then have the stomach to carry it out now that they know it is necessary to destroy you in order to save themselves.
Since political ideas over time survive primarily through the raising of new generations, one after another, over a very long time a difference in political ideas can lead to the creation of distinct ethnic groups. This split between people is far less stable than one caused by geographical boundaries, as the battle for hearts and minds requires continuous contact with the other group. After centuries of failed attempts by one side to dominate the other, and vice versa, the hardened feelings lead to the final endpoint of genocide.
The lesson here is that destroying an opposing political idea is a terrible path, as it can easily lead to your own destruction, and a worsening of the lives of your children, who are in fact representatives of your beliefs. It is far better to take the noble route by trying to convince others to come onto your side, and conversely be open to move to the other. More generally, the best strategy is to attempt to integrate opposing viewpoints into a system of mutual respect and discussion, where the methods and means to convince each other of the merit of your ideas are fair, balanced, and lead over all to a better future for both sides' progeny.
That means democracy, of course, where the EmbodiedIdeas of the masses are represented by votes. Votes are fundamentally symbolic stand-ins for which side of a hypothetical civil war we'd stand on if things got that way. Over time, though, vote after vote without a real threat of civil war, it becomes obvious that it is not really necessary to eliminate the bodies that hold unfavourable ideas most of the time, as most ideas are not so immediately threatening that a few years testing them out won't be worth the effort.