Here are links to the first three articles in this series: Introduction, Ad Hominem, Strawman.
This is probably one of the most common fallacies you'll come across. To beg the question is essentially assuming what you're trying to prove. But if you're trying to prove something, then assuming it implicitly in a premise of your argument won't be convincing, even though the argument is a valid one.
Philosopher Matthew Flanagan describes this fallacy as follows: "'Begging the question' refers to the informal fallacy known as petitio principii, which literally means 'requesting first principles.' The 'question' in 'begging the question' refers to the matter at the heart of the debate, the issue being debated. To 'beg the question' is to attempt to have that question conceded by assuming it either implicitly or explicitly in the premises of the argument the arguer offers for its truth. In other words, the arguer assumes what he [is] trying to prove and uses that assumption to prove that assumption correct."
Begging the question differs from other fallacies in that a question-begging argument is valid, whereas most other fallacious arguments are invalid. An example of a question-begging argument would be:
P1: God wrote the Bible.
P2: The Bible says that God can't lie.
P3: Therefore, God exists.
It's true that if God wrote the Bible and that God can't lie, then everything in the Bible is true, including the fact that God exists. But the problem is that you must first accept the premise that God exists before this argument will be convincing to you. Of course, there are independent reasons to believe that the Bible is divinely inspired, but this argument won't get you there.
Circular reasoning is a specific type of question-begging argument. Not all question-begging arguments are guilty of circular reasoning, but many are. An example of a case of circular reasoning would be something like the following: Women are better at writing poetry because men don't write poetry as good as women do.
Now, not everyone will make it easy on you to identify fallacies. Usually, unless someone specifically studies logic, the person you are talking to won't put their argument in the form of a syllogism. If you learn to do that, though, recognizing errors in reasoning will become much easier.
It also bears mentioning that people often confuse begging the question with raising the question, but the two are not interchangeable. For example, someone might say, "the weatherman this morning said there is a sixty percent chance of rain, which begs the question: Should I take an umbrella, just in case?" What this person really means is that it raises the question. It doesn't beg the question, which is a logical fallacy.
Examples of this fallacy in the abortion issue:
Many arguments that pro-choice people make beg the question. What is at the heart of the abortion issue is whether or not the unborn are full human persons (that is, human beings that have a serious right to life). So arguments from situations beg the question because you have to assume that the unborn are not valuable human beings with a serious right to life in order for the argument to succeed. So the argument that women need abortion because they might not be able to afford a child begs the question because we wouldn't allow a mother to kill her two-year-old child to make it easier to afford raising her other children (or because we think that growing up poor will mean the child won't have a good life). So if the unborn are full human persons, we can't justify abortion for that reason, either.
Pro-life people can beg the question, too, especially religious ones. The argument that abortion is wrong because God has commanded us not to murder is a question-begging argument. You have to assume that God exists for this argument to have force. But of course, most pro-choice people we talk to will not be religious, so this argument won't be convincing to them.
This is a fallacy that many of us make often without even realizing it. But as we learn to think more clearly and to give reasons for our views, we will stop making so many assumptions and be able to give compelling reasons for why we hold to the position that we do.