Anecdotal Fallacy

Anecdotal evidence is basically "This happened to me/someone I know/someone I heard about." Such evidence is largely useless as proof, since it is by nature doubtful in veracity. Furthermore, even if "it" did happen to someone the speaker knows, this does not establish that "it" is common or pervasive, which is the point the speaker is usually trying to argue. Typically used as the basis of a Hasty Generalisation, where a single, anecdotal "proof" is taken to prove or disprove a general rule.

"According to statistics, smoking causes you to die young. But my Grandmother Sally smoked like a chimney and lived until she was 95, so clearly the statistics are wrong."

As a rebuttal, one might simply point out that they met a man on the way home who said that anecdotal evidence doesn't prove anything.

Examples of Anecdotal Fallacy include:

A belief that some event was caused by the supernatural usually falls into this category. For example, if you asked God to heal your grandma, and she got better, that wouldn't prove God did it. While such an episode might be powerful to the individual, it is only a data point, and does not prove anything. A hundred other people might ask for the same thing, and get a dead grandma.
- Even better, people on both sides of the debate will pull this off. If your grandma didn't make a recovery then God doesn't exist, but you dismiss any cases were she did as 'lucky', while believers will take any recovery as unassailable proof of God, and any case without a recovery was God's will.

Looks like this fallacy but isn't

When the anecdote is offered as a data point, not proof.
When the anecdote is used to counter a hasty generalisation; using the example above, claiming that grandma is a long-lived heavy smoker with no health problems won't prove that smoking is healthy, but it can disprove the claim that "All smokers will get cancer".