The 1978 Nobel Prize in Physiology or Medicine was awarded to Werner Arber, Daniel Nathans and Hamilton O. Smith for their discovery of restriction enzymes. Restriction enzymes cut DNA at specific recognition sequences and are used to insert foreign DNA into a vector for cloning. The process involves cutting the DNA to be inserted and the vector with the same restriction enzyme, ligating them together, transforming bacteria, and using techniques like gel electrophoresis and antibiotic selection to identify recombinant plasmids containing the inserted DNA fragment.