DNA looks like a long double-stranded helix. One reason for this is for protection against mutation. If something happens to one of the strands of DNA, it is possible to read the opposite strand to make a new copy of the damaged strand. To simplify things, think of DNA as a long piece of ticker-tape. Written on this ticker tape are all sorts of instructions, or genes.
Genes are discrete units of information that tell cells what to do.
For example, there might be a gene for brown hair or a gene for green eyes. There are controls in place to make sure that the correct genes get expressed in the correct tissues - to ensure that you grow fingernails only on your fingers and hair only on specific parts on your skin, etc. Proteins are made by linking amino acids together. DNA contains the instructions, proteins carry out the instructions. So if DNA makes proteins and proteins perform functions, then the ultimate goal of genetic engineering is to alter proteins.