A kind of junk DNA
which does not benefit the rest of a genome
but which manages to survive and spread.
Junk DNA is any DNA that is not translated to proteins. But some of it might have a purpose anyway, and be coded for by other genes: for example it might physically bulk up the cell nucleus or space out true genes. Any such useful purpose would be controlled by other DNA. "Selfish" DNA is different, in that it is detrimental to other genes (it wastes space, consumes resources, but does nothing to aid their survival).
Its existence was predicted by Richard Dawkins in his 1976 book The Selfish Gene, almost in passing, as a consequence of his generalized theory of replicators being anything whatsoever that had the ability to aid their own survival and therefore undergo evolution. The name "selfish DNA" was coined as the title of a 1980 paper in Nature by Leslie Orgel and Francis Crick, who were one of two groups who simultaneously worked out the theory. It is also discussed in detail in Chapter 9 of Dawkins's The Extended Phenotype.
Among all the stretches of junk DNA, whether called into existence by the genome or arising by accidental mutation, some may have properties that make them harder to remove than others. Perhaps the molecular proofreading mechanism slips on a certain sequence of base pairs, either cannot see it as well or cannot grip it if it does detect it. Perhaps the sequence has just the right shape to hitch lifts efficiently when replication is going on. Perhaps it even mimics functional genes.
For whatever reason, if it has what it takes to be replicated, it will be replicated. It does not have alleles the way regular genes have, for it exists outside the meticulously policed system of chromosome duplication and cell-division.
Probably like most parasites, most selfish DNA is fairly harmless, because if it damaged its host it would suffer along with it. But an extremely dangerous kind of rogue DNA is a segregation distorter, one that increases its own chances above 50% of getting into the next generation. This is so dangerous because at the generation-by-generation level of genic selection it increases, regardless of its longer-term effect on fitness. It can't be weeded out by the slower process of natural selection, and a gene pool with a detrimental segregation distorter is likely to plunge to extinction in a very short time.
It may be that a considerable part of the cellular policing has evolved precisely as a firewall to keep selfish DNA from spreading.
It is quite different from the selfish gene. All successful genes are, by definition, selfish. But selfish DNA is a rogue as far as all the other DNA is concerned.