The term meme (pronounced to rhyme with "dream") was coined by Richard Dawkins in the book The Selfish Gene to mean roughly a unit of cultural evolution, analagous to gene, the unit of biological evolution. (The concept, however, predates the coining of the term; for example, William S. Burroughs' assertion that "Language is a virus"). Memes can represent parts of ideas, languages, tunes, designs, moral and esthetic values, skills, and anything else that is commonly learned and passed on to others. The study of memes is called memetics.
Dawkins observed that cultures can evolve in much the same way biological populations do, by passing ideas from one generation to the next, some of which may enhance or detract from the survival of the person holding them, thereby affecting which of those ideas continue to be passed on to future generations. For example, early cultures may have had different designs and methods for building tools. The culture with the more effective method may well have prospered while others suffered, leading to its method being adopted by a higher proportion of the population as time passed. Unlike biological genes which are usually "selected" by the death of competing organisms, memes can be selected--and therefore evolve--by more gentle means such as criticism, persuasion, and even "fashion".
Evolution requires not only inheritance and selection, but also mutation, and memes clearly have this property as well. Ideas that get passed on may undergo changes that accumulate over time. Folk tales and myths, for example, are often embellished in the retelling to make them more memorable--and therefore more likely to be retold again. More modern examples can be found in the various urban legends and hoaxes that circulate on the Internet, such as the Goodtimes virus warning.
In much the same way that the selfish gene concept can be used as a point of view from which to better understand and reason about biological evolution, the meme concept can be used to better understand some otherwise puzzling aspects of human culture (and learned behaviors of other animals as well). A controversial application of this is the idea that certain collections of memes can act as "memetic viruses": collections of ideas that behave like independent life forms, and continue to get passed on even at the expense of their hosts simply because they are good at getting passed on. It has been suggested that evangelical religions behave this way; by including the act of passing on their beliefs as a moral virtue, other beliefs of the religion also get passed along even if they aren't particularly valuable to the believer.
It is common for the term to be misused to refer specifically to these supposed "viral" meme collections, and to speak of a person being "infected" by such ideas. While this may be a useful rhetorical technique, such popular usage has somewhat tarnished the scientific respectability of the meme concept in general in fields where it has been used legitimately.
See also Copycat.