Stable transfection is the type of transfection where genetic material that is introduced into a cell is retained beyond reproduction. Transfection is the process in which foreign DNA is introduced into eukaryotic cells through chemically-created "gates" in the cell membrane.
In some transfections, DNA enters the cell nucleus and is added to or completely replaces the DNA of the eukaryotic cell. These are stable transfections. Only stable transfections allow the new DNA to be reproduced when the cell divides to create daughter cells. Under normal circumstances, it is difficult to produce a completely stable transfection; the tiny percentage of cells produced will soon die off and be replaced by cells with the original genetic material
Stable transfections that are experimentally useful are induced by co-transfecting another gene that can give the cell a selection advantage, typically resistance to a particular toxin. After mitosis, the cells produced are exposed to the toxin. The cells with the co-transfection -- and the transfection you originally wanted -- will survive, while most of the normal cells will die. After many rounds of mitosis and toxin, only cells with the resistance and the desired genetic change will survive.
Stable transfections are at the core of gene therapy. Only through a stable transfection can a faulty gene be permanently replaced.
Web Resources On Stable Transfection
Stable Transfection Protocols
Wikipedia: Gene Transfection
Book Resources On Stable Transfection
Gene Transfer to Animal Cells by Richard M. Twyman
Viral Vectors for Gene Therapy: Methods and Protocols by Constant & Machida