The term was coined by Richard Dawkins1. Just as genes are carriers of information that frame our make up, memes are carriers of information, of the idea variety. It is a term to talk about the spread of ideas, which in this era of bit space is a good deal easier and hence more rapid than in earlier times. Wikipedia has a more detailed account of the Meme.
The notion is not free of controversy, see, for example: Aunger, R. (Ed.) (2000). Darwinizing culture : the status of memetics as a science. Oxford ; New York: Oxford University Press.
This from the forward of the book by Daniel Dennett:
If there is one proposition that would-be memeticists agree on, it is that the flourishing of an idea – its success at replicating through a population of minds – and the value of an idea – its truth, its scientific or political or ethical excellence – are only contingently and imperfectly related. Good ideas can go extinct and bad ideas can infect whole societies. The future prospects of the meme idea are uncertain on both counts, and the point of this book is not to ensure that the meme meme flourishes, but to ensure that if it does, it ought to. It works towards this worthy end by creating a landmark, a fixed point not of doctrine but of evidence and methods, some shared acknowledgement among the leading proponents and critics about how the issues ought to be addressed.
Some other meme-related reading:
Shifman, L. (2011). An anatomy of a YouTube meme. New Media & Society. doi:10.1177/1461444811412160
Blackmore, S. J. (1999). The meme machine. Oxford England ; New York: Oxford University Press.
Aunger, R. (2002). The electric meme : a new theory of how we think. New York: Free Press.