Follow the following links to the different parts in this series: Introduction, Ad Hominem, Strawman, Begging the Question, Slippery Slope, and Equivocation.
The term "non sequitur" is a Latin term that simply means "it does not follow." A non sequitur is committed when an argument does not follow logically from its premises. This is obviously fallacious since in order for an argument to succeed, it must be both valid and sound (see the introduction for a refresher on the difference).
Let's take the following argument, that I found at this linked site: Maria drives a car. Maria must be rich. But it obviously doesn't follow from the fact alone that Maria drives a car that she must be rich. Perhaps she is poor but was given the car as a gift.
Here are a couple of examples of Non Sequiturs:
An argument a pro-life person might make would be: The only reason someone would be pro-choice is because they hate babies. But that doesn't follow at all: there may be some pro-choice people who hate babies, but one does not have to hate babies to be pro-choice. Many pro-choice people don't believe the unborn human being to be a baby, and some believe that it is a baby but that the mother's right to bodily autonomy trumps the child's right to life.
An argument a pro-choice person might make is: the unborn are not valuable because they cannot breathe on their own. But again, this doesn't follow. People on life support are still human, even though they can't breathe on their own (whether or not they have a good chance of recovery). The fact that someone is more dependent does not give us grounds to treat that person any way we wish. In fact, some might say that someone being more dependent gives us a greater obligation to care for that person.