Cognitive dissonance is a term that I often see misused, especially as I read (and engage in) online discussions. Cognitive dissonance is defined as "the mental stress or discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values at the same time; performs an action that is contradictory to one or more beliefs, ideas, or values; or is confronted by new information that conflicts with existing beliefs, ideas, or values" (Leon Festinger, A Theory of Cognitive Dissonance, California: Stanford University Press, 1957, as quoted by Wikipedia).
This article, written by Saul McLeod, summarizes the work of Leon Festinger. Festinger argued that the reason we experience cognitive dissonance is because we have a powerful motive to maintain cognitive consistency, which can give rise to irrational and sometimes maladaptive behavior. McLeod writes:
"According to Festinger, we hold many cognitions about the world and ourselves; when they clash, a discrepancy is evoked, resulting in a state of tension known as cognitive dissonance. As the experience of dissonance is unpleasant, we are motivated to reduce or eliminate it, and achieve consonance (i.e. agreement)."
Cognitive dissonance is not simply holding contradictory beliefs. I often see this term thrown around at someone who simply holds beliefs that are in contradiction. But there could be many reasons someone holds contradictory views. Perhaps they just don't care enough to examine their own beliefs. Or perhaps they aren't particularly trained in logic so they haven't been shown how their views contradict. Cognitive dissonance, though, is the psychological stress one feels at realizing that two views they hold, or new information presented to them, are contradictory and cannot both be true. This can result in different outcomes, such as: selectively exposing themselves to information that only favors their view, selectively retaining information that pertains to their view and losing information that conflicts with it, devaluing the new information and keeping their old view, or devaluing the old information and changing their views (which is the desired outcome, if you're attempting to persuade someone).
Think of a piano. If you walk up to a piano and press two notes that are a half step apart (C and C# work the best for this), it sounds grating, like nails on a chalkboard. This terrible sound is called a dissonance, and in composing music should be used very sparingly. The opposite is called consonance (when you play an interval or chord that sounds nice). This grating feeling you get when hearing a musical dissonance is comparable to the psychological stress someone feels when they realize they are holding contradictory views.
This is why asking questions is so important. By asking someone questions, you might be getting them to evaluate their own views for the first time so that they can see for themselves the inconsistencies in their own views. It leaves more of an impact if they come to the conclusion themselves rather than having you tell them. This happens fairly often at my pro-life outreaches. People rarely change their minds on the spot, but in time their cognitive dissonance may give them the push they need to reject their view and accept yours.
One of the most common ways to expose inconsistencies in a pro-choice person's views is by trotting out the toddler (a technique developed by Scott Klusendorf of Life Training Institute). Many pro-choice people believe that abortions are justified in certain situations, such as when the family is in poverty or the mother doesn't feel ready to raise a child. But they, of course, would never accept this reasoning if a mother wanted to kill her toddler. By pointing this out and asking questions, inserting a toddler into the equation instead of a human embryo or fetus, we can shine a spotlight on someone's inconsistencies and help them realize the contradiction for themselves.
Now, if a pro-choice person did not believe that the unborn are human persons (e.g. if she honestly believes the unborn entity is just a clump of cells or "blob of protoplasm"), then trotting out the toddler won't be helpful. You'll need to engage with her misunderstanding regarding human development. There will be no conflict, no cognitive dissonance, because her own views are consistent within her own mind; they are simply incorrect (i.e. they do not match what is actually true).
For this reason, cognitive dissonance can actually be a good thing -- cognitive dissonance is what gets someone to reevaluate their views and possibly change them. Incidentally, this is also why asking questions is so valuable. The person you are talking to may never have actually evaluated their own views before, so are simply unaware of the contradiction they are holding. Of course, if you're trying to persuade someone, you want them to devalue the old information and value the new information you are giving them so they will change their mind. There are things someone can do to alleviate their cognitive dissonance that don't result in changing their minds, but at the least getting them to understand the inconsistencies in their own view is the first step in persuading them toward yours.