Epidemiology is the study of human health in populations, rather than in individuals. It studies the causes of illness, how a particular disease is spread, and disease control. Epidemiology is, literally, the study of epidemics.
Epidemiology first became an issue when human populations began to gather together in towns and cities. Without some form of rudimentary epidemiology, controlling outbreaks of infectious diseases would have been impossible. Simple quarantining can contain some diseases that are transmitted from person to person, but more sophistication is required to stop the spread of diseases whose cause is not so obvious, such as cholera.
Epidemiology employs a wide array of scientific tools, from medicine and statistics to sociology and anthropology. Many diseases follow the flow of population migrations, so an understanding of how a migrant population moves with the seasons is critical to understanding the spread of a particular disease in that population. Epidemiology is concerned not only with the spread of diseases, but also with their cause -- if an outbreak of cholera can be traced back to a particular contaminated water source, for example, the disease can be effectively halted by shutting off access to that water source.
Sometimes the cause of a particular health problem can only be shown statistically -- here epidemiology has its hardest case to make. That smoking is a factor in a number of health problems has so far only been shown statistically, in large population studies. An individual case of lung cancer cannot be proved to have been caused by smoking, which unfortunately gives some intellectual wiggle room to those who choose not to believe that smoking is unhealthy.
Many health issues caused by environmental factors are only addressable through epidemiology. A spike in the incidence of cancer near a source of pollution, for example, can not be seen when looking at individual cases, but only when looking at the statistics for the entire area. Epidemiology is the main weapon in the field of public health, and epidemiologists are often our first and last defense.