Although not strictly logical fallacy, confirmation bias is simply the tendency for individuals to favor information or data that support their beliefs. It is also the tendency for people to only seek out information that supports their a priori, or pre-existing, conclusions, and subsequently ignores evidence that might refute that pre-existing conclusion.
Technically, confirmation bias is a type of cognitive bias and a form of selection bias, which seeks data that confirms the hypothesis under study. It is utilized in arguments all the time, especially by those who do not accept or understand scientific methods.
Avoiding confirmation bias is an important part of rationalism. The scientific method, itself, was developed to remove biases. In science, it is achieved by setting up problems so that you must find ways of disproving your hypothesis (see falsifiability).
Example of confirmation bias
Anti-vaxxer — “There is an article by Dr. Smith and Dr. Jones that states that confirms that Andrew Wakefield was right about vaccines and autism.”
Pro-science/pro-vaxxer — “But there are hundreds of articles from research across the world published in respected journals that say autism is absolutely unrelated to vaccines.”
Anti-vaxxer — “Not relevant, only the article by Smith and Jones is important to this discussion.”