An anecdote is essentially a short story about an interesting or relevant event that happened to someone. Anecdotal evidence is when, rather than providing evidence based on scientific research, statistics, or experimentation, personal stories related to a similar topic are collected and offered as evidence.
An example of this would be a friend of yours discouraging you from patronizing a local grocery store by telling you the story of how they thought they were treated rudely by the manager when reporting on an employee not rushing to wash their hands after coughing on them. Though usually compelling (it can be hard to disregard a good story), anecdotal evidence is not necessarily the most reliable source of information.