There are a number of techniques for moving genes artificially into recipient organisms. The oldest of these is called recombinant DNA, a technique that relies on biological vectors like plasmids or viruses. Other newer gene transfer techniques are electro- and chemical poration, microinjection, and bioballistics.
Recombinant DNA techniques use biological vectors like plasmids and viruses to carry foreign genes into cells. Plasmids are small circular pieces of genetic material found in bacteria that have the ability to cross species boundaries. The circles can be broken and new genetic material added to them. Plasmids augmented with new genetic material can move across microbial cell boundaries and place the new genetic material next to the bacterium's own genes. Often the bacteria will take up the gene and begin to produce the protein for which the gene codes. Where the new gene codes for insulin, for example, the bacterium will begin to produce insulin along with its other gene products. A large vat of bacteria engineered to produce insulin can then become a sort of pharmaceutical factory.
Viruses can also act as vectors in genetic engineering. Viruses are infectious particles that contain genetic material to which a new gene can be added. The virus can carry the new gene into a recipient cell in the process of infecting that cell. The virus can also be disabled so that while it can carry a new gene into a cell, it cannot redirect the cell's genetic machines to make thousands of copies of itself.
Other methods do not rely on biological vectors like plasmids and viruses. One of these is called microinjection and involves simply injecting genetic material containing the new gene into the recipient cell. Where the cell is large enough, as many plant and animal cells are, the injection can be done with a fine-tipped glass needle. Somehow the injected genes find the host cell genes and incorporate themselves among them. Electro- and Chemical Poration Other methods for direct gene transfer involve creating pores or holes in the cell membrane to allow entry of the new genes. This can be done by bathing cells in solutions of special chemicals--so-called chemical poration--or subjecting cells to a weak electric current--so-called electroporation.
Finally, there are so-called projectile methods that use metal slivers to deliver the genetic material to the interior of the cell. The small slivers (much smaller than the diameter of the target cell) are coated with genetic material. One projectile method, called bioballistics, propels the coated slivers into the cell using a shot gun. A perforated metal plate stops the shell cartridge, but allows the slivers to pass through and into the living cells on the other side. Once in the cell, the genetic material is transported to the nucleus where it is incorporated among the host genes.
Source: The above information on genetic engineering techniques is from the Union of Concerned Scientists website.