Molecular biology is a field of biology that looks at the molecular machinery of life. The field was founded in the early 1930s, though the phrase was only used in 1938 and the field didn't take off until the late 50s and early 60s. Since then, progress in the field has been massive. The field began with the x-ray crystallography of various important biological molecules. Now, crystallography databases store the molecular structure of tens of thousands of these molecules. Understanding of these proteins both helps us understand how the body works and how to fix it when it breaks down.
Truly modern molecular biology emerged with the uncovering of the structure of DNA in the 1960s and concurrent advances in biochemistry and genetics. Molecular biology is one of three primary molecular-scale biological sciences, the others being biochemistry and genetics. There is no clear division between the three, but they do have general domains.
Broadly speaking, biochemistry looks at the function of proteins within the body, genetics looks at how genes are inherited and propagated, and molecular biology looks at the process of replication, transcription and translation of genes. Molecular biology has some surface similarities with computer science, because genes can be looked at as a discrete code, though the proteins they code for and their subsequent interactions can be highly nonlinear.
The most important idea in molecular biology is the so-called "central dogma" of molecular biology, which states that information flow in organisms follows a one-way street -- genes are transcribed into RNA and RNA is translated into proteins. Though generally correct, the "central dogma" is not as absolute or certain as its name implies. In some cases, information flow can reverse, as the protein environment can influence which genes are transcribed into RNA and which RNA is translated into proteins. The broad picture does hold, however, as if proteins had too much of an influence over the genes coding for them, the body would be in chaos.
One of the most basic areas of inquiry in molecular biology is the use of expression cloning to see which proteins are created by which genes. Expression cloning involves cloning a DNA segment coding for a protein of interest, attaching the DNA to a plasmid vector, then introducing the vector to another plant or animal. The way the transferred DNA is expressed provides valuable insights into its role in the organism. This allows us to learn what genes do. Without this knowledge, much of genetics, such as our knowledge of the human genome, would be useless.
There are many other lines of inquiry in molecular biology. The field is mind-bogglingly huge. The information presented above, however, serves as an introduction.