Genetic engineering refers to identifying, isolating, inserting and transferring the gene of interest from one organism to another organism. The organism receiving the gene of interest may be of the same or different species. The recipient organism would express the transferred gene.
The gene of interest has to be inserted into a DNA vector. The DNA vector discussed here would be a plasmid. The plasmid vector is DNA obtained from a bacteria.
Transgenic organism refers to an organism that has acquired a foreign gene.