What is the two alternative forced choice paradigm Is a two alternative forced choice paradigm (2AFC) an experimental design? 
 A: Two-alternative forced choice is a psychophysical method ("paradigm") used to solicit behavioral or perceptual responses from subjects.
The name comes from two aspects of the method. It is called two-alternative because the subjects are shown two alternative stimuli (A or B), and forced choice because they are "forced" to choose one, based on some previously-instructed criteria (e.g., brightness, moving rightward, etc). It is "forced" in the sense that the subject isn't offered a way to avoid answering or choose "none of the above"; it's not actually coercive.
The term is sometimes used more loosely to indicate a task where the subject views a single stimulus and must make one of two responses. However, this isn't strictly correct. Macmillin and Creelman (2005) (pg 166) write:

"In using the traditional name for this design, we continue an unfortunate historical precedent. The choices made by observer in two-alternative forced choice (2AFC) studies are no more constrained than in other correspondence experiments. As in the one-interval design, the possible stimuli come from one of two categories (Old or New, Loud or Soft) and the experimenter is interested in the correspondence between the correct response and the observer's 'forced choice.' The new feature of the 2AFC design is that both alternatives are presented on every trial in random spatial or temporal order. The observer reports not which stimulus occurred--both did--but in which order."

Most other sources seem to agree with this definition: Hautus, 2015;  Schneider and Parker, 2013.
This distinction is important because a "classic" 2AFC task is much easier than a detection task, and signal detection theory results about performance on one do not always apply for the other.
Consider Jeff's lexical decision task, below. In a detection paradigm, you might be shown the string "goke" and asked to decide whether it is a word (no). In the corresponding 2AFC experiment, you might instead be shown "goke" and "joke" and asked to decide which is a word (joke). Macmillin and Creelman present this a bit oddly, but you could also reply by indicating the relative order ("The word came 2nd").
The detection paradigm relies only on your knowledge of the string "goke", while the 2AFC task lets you weigh the (un)familiarity of "goke" against the familiarity of "joke", which provides the observer with extra information. In fact, it can be shown mathematically that an optimal strategy does a factor of $\sqrt{2}$ better on the 2AFC task.
As Jeff notes, some fields are less rigorous about this than others. Still, it costs almost nothing (two extra characters!) to describe a task in a way that everyone will unambiguously find "correct" (2AFC vs. yes/no), so I would encourage you to do so.
A: Two-alternative forced choice (2AFC) refers to the methodology used to solicit responses from participants. It is used in many different paradigms across different domains. As the name implies, participants are presented only two choices, and they must make a response (indifference is not allowed).
For example, the lexical decision task is a 2AFC task. Participants are presented a sequence of letters such as BOKE and must quickly indicate whether the sequence of letters forms a word or a non-word.
Another example from psychophysics might ask participants to judge whether a test stimulus is brighter than a target stimulus. Participants must response either yes or no.
A: Having read the other two answers, I think we are still missing an important point.  As @Jeff notes, many tasks involve two alternatives, e.g., a detection task. In a detection task, or indeed, in any task with a single stimulus and a decision that must be made about that stimulus, the subject must set a criterion level for their decision. This criterion level may vary from person to person, or may vary in the same person from day to day or even from trial to trial. In contrast, a task with two stimuli eliminates the possibility of different criterion levels. With fewer degrees of freedom to the subjects behavior, the experiment is more controlled.  (There may be bias in the 2 stimulus task - you might be biased to pick the stimulus on the left, or the stimulus presented first. This bias is orthogonal to the task, however, whereas criterion level directly affects task performance.)
