We're making arguments all the time. The purpose of an argument is to convince someone of something. It's important to make sure that all of the statements in the argument logically flow to the conclusion. Unfortunately, not all arguments do this well. Sometimes the conclusion can still be false, even if the main premises are true. This happens when the argument is fallacious (wrong).

There are many different logical fallacies, Here's a good article on common political fallacies. Here are some of the more common ones.

Ad hominem — Attacks the person, not the argument. “We can't trust his argument because he was arrested last year”. So what if he was arrested? That has nothing to do with whether his argument is right or not.

Ad ignorantiam — It's true because we don't know that it's not true. This is a popular one for Intelligent Design. Just because scientists don't know how a certain thing happened, doesn't mean that God did it.

Ad populum — It's true because lots of people think it's true. Everyone used to think that the sun orbited the Earth, but it was still wrong.

Argument from antiquity — It's true because we've always believed this. Just because a belief has been around for a long time doesn't make it true.

Argument from authority — Albert Einstein said that this is true. We all know how brilliant he was, so it must be true. Just because someone is an expert in one field doesn't mean that they should be considered an authority in another field.

Argument from personal incredulity — I can't explain it, so it must be false. Another one for Intelligent Design. Just because you don't understand Evolution doesn't make it false.

Begging the question — Having the question include the conclusion. “Have you stopped beating your wife yet?” This question includes the assumption that I have been beating my wife (which isn't true at all).

False dichotomy — Reducing the set of possibilities to only two. This oversimplifies a continuum to only two choices. Typically used to try to force your opponent to an extreme position. “If you're not with us, you're against us.” But being neutral is another option.

No true Scotsman — Trying to protect a generalization from counterexamples. Person A: “No Scotsman puts sugar on his porridge.” Person B: “My uncle Angus (a scotsman) likes sugar on his porridge.” Person A: “Ah, but no true scotsman puts sugar on his porridge.”

Slippery slope — A position shouldn't be taken because accepting it means the extreme position must be accepted. “If we legalize playing Call of Duty publicly, that will lead to playing it with real weapons, and then people will die. No one wants that.” Just because you play Call of Duty doesn't mean that you'll start playing with real weapons.

Special pleading — Adding new elements into an argument to make it appear valid. It can be difficult to detect. For example: ESP hasn't been observed in laboratory conditions. Defenders of ESP claim that ESP doesn't work when there are skeptical people around.

Strawman — A strawman argument is a logical fallacy where a person sets up a flawed position and then refuting it. This is called “attacking a straw man”. The flawed position that is setup is not the one that the opponent is defending. This creates the illusion of having defeated an opponent's position by replacing it with a different position. Person A: “I think more of our tax dollars should be spent on education.” Person B: “You hate our country and want to leave it defenceless by cutting military spending!” Person A didn't say anything about military spending, that was the strawman.

The moving goalpost — Changing the requirements for “proof”. Person A: “Show me just one intermediate fossil to fill this gap of time.” Person B: “Here's one.” Person A: “Now there are two gaps of time! You need to show more fossils!”. Person B has already provided a counterexample. Person A is “moving the goalposts” by demanding more.