A logical fallacy is a mistake in a person's reasoning. More specifically, the kind of explanation they are providing in support of their conclusion is flawed and will not work in either all or the vast majority of circumstances. In the case of the conjunction, any two events capable of occurring distinctly have a higher probability of both occurring individually than together. For instance, it's more likely an unknown person is a clown than a clown and a baseball star. This is because the probability of being a clown and a baseball star are both low for any given individual, and guessing correctly twice, so to speak, is always more difficult than to do so once.
The most famous example is by Amos Tversky and Daniel Kahneman. It is quoted as follows:
"Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations."
Those involved were asked which was more probable:
1. Linda is a bank teller.
2. Linda is a bank teller and a feminist.
After receiving this question, 85% of participants in the study thought it was more likely "Linda is a bank teller and a feminist" than "Linda is a bank teller." Again, even if it's extremely likely Linda is a both of these things, it's more probably she is a bank teller. The reason is clearer when you consider the following insult: "you are unintelligent and a human being." You're probably not unintelligent, but it doesn't mean you're not a human being. Similarly, Linda can not be "a bank teller and a feminist" without not being a bank teller. If #2 was true, that means #1 is also true. The probability of Linda being a bank teller includes the chance she will be a bank teller and not a feminist as well as the chance of her being both. By default, it's more probable.
The experiment has been studied further, and people seem less likely to make the error when they converse with others. Furthermore, the question may play on the way people speak in everyday terms. Sometimes people use sentences like "she is a doctor and a board member at the hospital." If she is a board member, she may be required to be a doctor. In that sense, the fallacy only applies to "and" questions where the two things are unrelated. Additionally, the use of conjunctions is more common in logic than everyday communication. Many people would not associate #2 with also making #1 true by default. This is a convention in a variety of fields more so than in social situations.
Tversky, A. and Kahneman, D. (October 1983). "Extension versus intuitive reasoning: The conjunction fallacy in probability judgment". Psychological Review 90 (4): 293–315. doi:10.1037/0033-295X.90.4.293. http://content2.apa.org/journals/rev/90/4/293.