A logical fallacy is an approach to making an argument or point by breaking rules that are known to not be sound in reasoning and logic. There are many many of these fallacies but I'll list and describe some of the main ones.
Ad Hominem (the argument against the man)
Attacking the person's character instead of attacking the argument.
"Why should I listen to your opinion on philosophy, you barely graduated high school."
Post Hoc Ergo Procter Hoc (After this, therefore, because of this)
This one is a favorite of mine. Drawing a conclusion that two or more events are related when there is no real proof they are.
"We ate fish tacos last night and this morning my head was pounding. Fish tacos give me headaches I guess" (the person isn't qualified to conclude fish tacos were definitely the cause of the headache)
"I had a dream I won the lotto, so I bought 20 scratch off tickets and one ended up being a winner for 500 bucks! I must be psychic."
circular logic (The conclusion of the argument is the same as the premise)
"The bible is the word of God because it says so in the bible."
Ad baculum (appeal to the stick or force)
An argument where force, coercion, or the threat of force is used to justify the conclusion.
"If you don't repent your sins you will burn for eternity in hell, therefore you MUST repent."
"Don't argue with the king's policies or he will lock you up and toss away the key. Therefore keep your mouth shut."
Ad lapidem (throwing stones)
Dismissing a statement as absurd without giving reason why it is absurd.
"You don't believe in a divine being? That's just fuckin' retarded!"
"You don't think marijuana is addictive? Are you crazy!? Pull your head out of your ass."
Argument based on assertions that cannot be tested.
"You're not old enough to really understand life."
"Eating beans prouts may not kill you today, but one day you may eat a poisonous one and die. Therefore you should just avoid eating bean sprouts."
"All atheists secretly believe in a god."
An attempt to divert the argument or change the subject
Person A- "Bush should have been tried as a war criminal."
Person B- "What about all the other people in the world that commit crimes and get away with it? What about that justice not served? In fact, the murder rate in Romania is bla bla yack yack....."
Asking a question that presupposes something that isn't yet proven.
"Do you still beat children up with baseball bats?"
"Why is Obama afraid to admit he was born in Kenya?"
Misrepresenting a persons argument in such a way it would be easy to refute. Beating up the straw man.
"Of course evolution isn't true. No one has ever seen a reptile turn into a bird."
"Liberals just want a big nanny state."
Ad Populum (appeal to the people)
Concluding an argument is true because a majority of people believe it to be true.
"Most people in the US believe marijuana is a dangerous drug therefore it is."
"Majority of the world believes there is a divine being, they can't all be wrong."
Using the fact that one wants to believe something to be true as evidence of it's truth.
"We are all beings of spirit and light."
Person A-"We are an electro-chemical system."
Person B-"That may be true to you, but that's not true to me. That's just your opinion."
negative proof (argument from ignorance)
Appealing to lack of proof of the negative. X is true because there is no proof X is false.
"There is no evidence Glenn Beck didn't rape and kill a woman in 1984."
"You can't prove god doesn't exist!"
Argumentation ad misericordiam - (appeal to pity) Recommended by Dryice
Using an emotion like pity, sympathy, or compassion for the sake of getting a conclusion accepted.
"If you don't give me a job here, I can't donate to the starving children's fund! My life dream would be ruined."
"Of course the judges should vote me winner of the chili cookoff, I've recently had a death in the family."
false dilemma or false dichotomy - (either-or fallacy) Recommended by tongues
An argument where only two choices are given when there are in fact more.
"You're either with us or with the terrorists."
"Either love your country with all it's problems or leave it."
inductive fallacy - (Hasty Generalization) Recommended by mrgoodsmoke
Fallacy committed when one comes to a quick conclusion about a population based on a sample that is not large enough.
"My town is mostly republican. Three of the five houses on my street have republican signs in them."
"I did a survey in my town by going to three different Dentists and they all said Crest was the best toothpaste to use so Crest must be the best."