The term "selfish gene" was coined by Richard Dawkins in his controversial book of the same name as a provocative way of expressing a particular point of view from which to look at the processes of evolution: that of imagining genes (rather than organisms or species) as the primary drivers and beneficiaries of the evolutionary process. More precisely, an organism is expected to evolve to maximise its inclusive fitness--the number of copies of its genes passed on globally (rather than by a particular individual).
Describing genes with the term "selfish" is not meant to imply that they have actual motives or will--only that their effects can be described as if they do.
A crude analogy can be found in the old joke "A chicken is just an egg's way of making more eggs." Likewise, Dawkins describes biological organisms as "vehicles" used by their genes for making more copies of those genes, regardless of the effect they might have on individuals or species. Obviously, genes that tend to help the organisms they are in to survive and reproduce also help themselves; so most of the time the gene and the organism share common goals. But there are exceptions: segregation distortion genes, for example, that are detrimental to their host nonetheless propagate themselves at its expense. Likewise, the existence of junk DNA that provides no benefit to its host, once a puzzle, can be more easily explained.
Proponents argue that the central point of the idea, that the gene is the unit of selection, is a more accurate recasting of evolution than Darwin's (who couldn't have explained natural selection in these terms because the basic mechanisms of genetics weren't understood at the time). Critics argue that this view oversimplifies the relationship between genes and the organism.
Proponents argue that when looked at from the point of view of gene-selection, many biological phenomena that were difficult to explain in terms of prior models of evolution become easier to understand and explain. In particular, phenomena such as kin selection and eusociality, where organisms act against their individual interests (in the sense of health, safety or personal reproduction) to help related organisms reproduce, can be explained as genes helping copies of themselves in other bodies to replicate. In other words, genes act "selfishly" to increase the number of copies of themselves and for no other reason.
The idea is sometimes mistakenly believed to support genetic determinism. Dawkins is quick to point out that although we may be influenced by our genes, we are not controlled by them. Even further from Dawkins's conception is the misunderstanding of the idea as predicting (or even prescribing or justifying) that human behaviour must be "selfish" in some sense.