Perhaps a useful analogy would be a recipe. Consider the DNA to be the overall recipe. The genes are the instructions. It is the DNA that contains the information (the genes) needed to produce proteins and RNA molecules, all of which are needed for life to exist.
DNA encodes all genetic information that allows cells to produce proteins. Genes are the parts of the DNA that actually encodes the proteins, as opposed to simply coding for nonsense or acting as a structural element.
DNA is the holder of genes. DNA is double helical in shape and composed of four different chemicals. Genes are composed of groups of these chemicals--a subset of the helix. It's amazing four chemicals can control such complex living beings.
DNA is the building block of genetic material. Genes are the aggregate expression of what your DNA will produce. You can think of it as DNA being the steel and rubber used for your car, which would be genes.
Genes are the common name for the genetic material that is encrypted in DNA. DNA is used by the body to build proteins, which work together to determine the characteristics that make each individual unique. A "gene" is the coding for a particular characteristic, or a particular protein. DNA is the coding for all of them.