Genetic engineering is a biotechnology technique where DNA from one organism is transferred to another organism. The reason we use this technique is so the organism that receives DNA becomes genetically modified, and will contain genes that give them physical characteristics which are useful to us. To use an example, let's talk about producing the hormone insulin through genetic engineering.
To start, we would take a human cell and identify the gene for making insulin. We would use an enzyme (called restriction enzymes) to remove this gene from the human chromosome. Then, we would take plasmids (small circular pieces of DNA found in bacteria) from bacterial cells and cut them open with another restriction enzyme. We would then place the gene for insulin inside these plasmids, and seal the plasmid up using another enzyme called DNA ligase. If the plasmid 'takes up' the DNA, it is said to be transformed and will start producing the hormone insulin, because it contains the genes needed to do this.