A transgenic organism is a genetically engineered organism. They are created by isolating DNA and then fragmenting it using restriction enzymes, before recombining it in a new, controlled way. The restriction enzymes cleave the DNA at a specific restriction site and mainly create sticky ends that have complementary sequences to each other and can therefore adhere together.
The recombined DNA is reintroduced to a cell or organism. The main use of this technology is for experimental studies, however a very important aspect of the application is developing and improving varities of crop plants and domesticated animals.
Transgenic mice are often used to study human disease responses as their genes can be modified to express the same mutations. New treatments can then be developed and trialled on the mice as they would respond in the same way as humans.