GMOs, or Genetically Modified Organisms, are created by transferring a gene from one species to another. Because all living organisms have DNA, it is possible to clip a section of DNA from one organism and insert it into another organism and have it work in its new location.
While doing this is theoretically straightforward, the actual technology is fairly complex. First you must identify the exact boundaries of the desired piece of DNA in the donor organism. Then you must create the correct restriction enzyme, which is a protein that will cut the DNA at a specific point; this restriction enzyme is used like a chemical scissor to cut the desired gene out, leaving the rest of the donor organism's genes behind.
The isolated donor gene is then inserted into a plasmid, which is a ring of DNA normally found in bacteria. The modified plasmid can then be used to make many copies of the DNA, a process known as amplification. Then the donor gene is introduced into the recipient organism. This is commonly done by inserting the donor gene into a modified virus, and letting the virus inject the gene into the target cells. In the case of plants, or tissues, any cell culture can be used for target cells. If you are creating a GMO animal, the target cell is usually a fertilized egg cell, which must then be inserted into a surrogate mother and allowed to develop naturally.