What's the difference between genes and DNA?

Genes are actually a subset of a cell's DNA. While all of your genes are made of DNA, your entire DNA is not composed of genes. In fact, less than two percent of a person's DNA represents active genes! The rest of the DNA seems to be involved mediating how the genes are expressed.
DNA, deoxyribonucleic acid, is found as long chains, with each "link" called a nucleotide. The structure of DNA is the well known double helix. Each bacterial cell generally contains a single chain of duplex DNA, called a chromosome, with about five million links in it. By comparison, cells in human beings contain 2 copies of 23 different chromosomes with around 100 million nucleotides each.
Genes were classically defined as the fundamental units of inheritance. Today we understand genes to be portions of DNA that contain the information needed by cells to live. In particular, genes are special sequences of nucleotides that are used to design proteins which carry out the work of building, maintaining, and reproducing the cell.
One can think of a genome, the sum total of all an organism's DNA sequence, as a book written in a special code. In this analogy, nucleotides are the dots and dashes of the code. Some pages of the book have instructions on how to make different proteins, these would be the genes. The other pages may have messages telling the cell where to begin making new DNA (origins of replication), how to read and edit other messages (promoters, terminators, and splicing signals), where to leave bookmarks for ready reference (binding sites), or where to bind the book (centrometric and telomeric regions)