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Affirming the consequent:

This claim is most simply put as

If A, then B.
B.
Therefore, A.

It's a fallacy because at no point is it shown that A is the only possible cause of B; therefore, even if B is true, A can still be false. For example

If my car was Ferrari, it would be able to travel at over a hundred miles per hour.
I clocked my car at 101 miles per hour.
Therefore, my car is a Ferrari.

This is popular in conspiracy theories. Here the fallacy is fairly obvious; given the evidence, the car might be a Ferrari, but it might also be a Bugatti, Lamborghini, or any other model of performance car, since the ability to travel that fast is not unique to Ferraris. Hell, it might even be a Subaru Outback. Note that while this may appear to call all hypothesis / evidence experiments fallacious, they are based on additional evaluations of the likelihood of other theories, thus establishing that A is a likely cause of B.

Denying the antecedent:

The flip side of the above, where you say that because the initial conditions did not happen, the result is impossible.

If a person is wearing a hat, they have a head.
I am not wearing a hat.
Therefore I do not have a head.

Note that, by the contrapositive rule, these two fallacies are equivalent. For example, you could replace "If a person is wearing a hat, they have a head" by the logically identical statement "If a person has no head, they aren't wearing a hat" to turn the first example of denying the antecedent into an example of affirming the consequent.

Examples:

• In American Dad, Stan sinks his entire savings to build a rocket for Steve to win a contest.
  Stan: You gotta spend money to make money.
  Francine: But you didn't make any money!
  Stan: So logically, I didn't spend any money! *waves at the camera* Goodnight everybody!

• A current argument made by Obama supporters against conservatives.
  Racists who don't like black people oppose Obama's presidency
  Bob opposes Obama's presidency
  Therefore Bob is a racist.
  • This is not to say that you can't make an argument that someone that opposes Obama is a racist, but it does not follow automatically from being opposed to his presidency and/or policies.
  • A similar argument from Obama detractors is that anyone who voted for Obama did so only for affirmative action's sake, rather than because they believed Obama was a strong candidate on his own merit.
  • I opposed Obama because I was for Hillary, you sexist!

Looks like this fallacy but is not:

• Inference to the best explanation. The usual form of scientific reasoning, as well as a lot of Sherlock Holmes' "deductions" (though he's wrong to call them that, since this is a form of inductive reasoning).
  B.
  The best explanation for B would be A.
  Therefore, A (probably).
  • This differs from the Ferrari example above in that it posits a stronger connection between A and B than just A's truth entailing B's; B is actually giving some positive reason to prefer A over the other possibilities. Also, this form of argument isn't claiming deductive certainty, so the bar is a little lower for it being acceptable.
  • Scientific reasoning is frequently attacked by those who understand this fallacy, but not the scientific method, which has the following form:
  B.
  A is the best explanation for B, so I will claim "A is the most likely explanation."
  If A, then C.
  Therefore, if not C, not A (valid contrapositive).
  Is C true? Yes? I will increase my confidence that A is the correct explanation.
  If A, then D.
  Not D!
  I must provisionally reject A or modify it to account for D, then continue to seek new information and propose new possible explanations.