A logical fallacy is an argument that contains a flaw in its reasoning. Some examples of common logical fallacies include an appeal to authority (when you argue that because someone in authority says something is true it must be), and begging the question (where the conclusion is included in the premise, eg. I know that the Flying Spaghetti Monster is real because the Holy Book of Pastafarianism that he dictated to his prophet says that he is real).
I can't say I have a favourite but I'm sure most of us as children were subjected to the appeal to emotion fallacy, "Eat your vegetables, think about all those poor starving children in Africa," where the eating of vegetables in one country has impact on the lack of supply in another.
In 2013 our (Australian) Prime Minister Tony Abbott came under much criticism for his non sequitur logical fallacy regarding the effect climate change had on bushfires when he claimed that because Austtralia has had bushfires since the beginning of time we cannot say they are impacted by climate change. The Guardian likened this to claiming guns can't kill people because people were also killed before guns were invented.