In my interactions with non-scientists there are some common misunderstandings about how science works. These misunderstandings are predominantly the result of a lack of exposure to science. At school science is offered as a subject but students rarely 'do' science. They get taught a bunch of facts and occasionally do some experiments with known outcomes. Realistically this is hardly science. Science is a way of approaching problems, questioning areas surrounding these problems and applying reason to get some understanding of the problem so we can begin to explain it.
Admittedly this procedure may be somewhat difficult to do at the high school level and indeed even at the tertiary level - at least it seems that way based on my very limited experience with teaching tertiary students. Nonetheless, certainly it can be achieved at tertiary and secondary level. I know that at the university I attended there has been some development in that area for first year students of science. The problem, however, is that science students are the only people with any semi-useful level of scientific literacy. The media reports on science on a daily basis but people are not qualified to make appropriate judgements on the science. What is worse is that the media always adds some slant to the reporting that often influences people into making up their mind about something with out actually being informed about it. I have written about a particular case in a previous post where the media got it quite wrong on the underlying scientific context that resulted in scores of ignorant comments on the news story.
So what is the answer? I don't have a clear-cut one but I have some possible ideas that might help. One of the misunderstandings that appears common - actually is is more of an ignorance - is how science is disseminated. It is very rare that I talk to people about what I do that have any clue about what the scientific literature is let alone how it works. I have found that when I tell people what it means to publish work in a journal they are surprised that it is common for work to be rejected for publication and, that many more manuscripts are rejected than accepted. They are also surprised that journals can charge you to publish, not the other way around.
Because publication is generally the final step in a research project (Note: How the knowledge gained from the project is a different discussion) I think that the public should know more about how it actually works, specifically peer-review. Peer-review is science's quality control and I think if more people understood the process they would be able to better gauge the merits of science. It might stop ridiculous phrases like "so-called scientists" when people disagree with the outcome of a study like the one I mentioned above. Rather they might realise that it is not just some persons pet idea that they have come up with, but actually something that has been scrutinised by other experts. When the work is controversial it will most likely be heavily scrutinised. It is of course important to admit that peer-review is not flawless, something I have briefly touched on before, but importantly this is the exception not the rule or anywhere near it. The University of California at Berkley has a good explanation of the peer-review process up on their website which I recommend for a brief introduction.