If you are certain that you are right:
Against a rational: Treat it like trapping a dangerous animal. Slowly nibble around the edges of the argument, reducing the amount of room they have to move until they are forced to admit defeat.
Against a non-rational: Engage them in discussion to determine what has convinced them of their opinion. It helps to understand their underlying value system. Make a convincing argument aimed at fitting into their value system.
Against someone incapable of rational discourse: Drugs and electroshock therapy.
If you are unsure of your position or it is a grey area argument:
Against a rational: Try to be more structured in your approach than your opponent, avoid bringing attention to weak points in your argument. Try to shore up your argument from multiple angles.
Against a non-rational: Essentially the same as before. You will probably have to counter more rebuttal. How effectively you do this is essential to winning.
Against someone incapable of rational discourse: Probably not worth the effort.
If it is a formal debate:
It is all about structured argument optimized for clarity and balanced rational/emotional appeal.