DNA is a long, flexible, double helical molecule found in all the cells of all the living organisms. It is made of four subunits, Adenine, Thymine, Cytosine and Guanine. These subunits, called nucleotides, are the rungs of the ladder in the diagram to the right and are arranged in a specific sequence that often code for the production of proteins. However, these sequences not only code for the production of proteins, but for the production of protein-regulating factors. This is the part we are interested in! We want to understand how the sequence of the DNA can determine when and how much a given protein is produced by the cell.
Proteins are also made of subunits called amino acids that form relatively strong bonds with one another called peptide bonds. These strings of molecules adopt three-dimensional structures that determine their function. You can see nice examples of proteins in interactive 3D in the pages of Proteopedia or The Protein Data Bank 101 where I found this nice video about proteins and their structure.
DNA in the cell is never alone, it is always found bound by proteins. These proteins can package it, protect it, repair it from damage, copy it when the cells divide, and regulate the expression of the genes, coding for other proteins. In bacterial cells the protein-DNA complexes tend to be very dynamic, with proteins coming and going all the time from a given site on the DNA. On the other hand eukaryotic cells have a more stable organization of the chromosomes, especially in specific "genetically silent" regions called heterochormatin.
Each cell needs DNA. When a cell divides in two it needs to copy its DNA so that each daughter will have a full genome. This is the process of DNA replication.
Below are two images from Wikipedia showing the protein Cro bound to the DNA. In the first image, the protein is represented either in a space filling model (below), which more accurately demonstrates the protein's real volume, or with a model that shows the path of the amino acid chain (above). The negatively charged DNA becomes slightly distorted in its attempts to wrap around the positively charged protein.
In the second image, the DNA is more dramatically distorted. This happens when the DNA wraps around organizational proteins called histones. This DNA-protein complex is called nucleosome and provides DNA with the ability to package and condense genetic information into chromosomes.