Criminology is the study of crime. More accurately, it is the study of crime as a social trend--its overall origins, its various manifestations and its impact upon society as a whole. That makes it more a form of sociology than a law enforcement tool. But the trends it studies have a huge impact on the way the police do their jobs, the way society treats its criminals, and the way a given community goes about maintaining law and order.
Modern criminology is usually traced to Cesare Lombroso, a nineteenth century Italian doctor who treated criminal patients in prison. He developed a positivist theory of criminology which stated that some criminals are biologically predisposed towards crime. This ran counter to earlier thoughts on criminology, which held that people are basically decent and driven to crime only through extenuating circumstances, such as poverty. Lombroso was revelatory because he recognized distinctions in motivations, which inferred that methods other than punishment could be used to reduce crime.
French sociologist Emile Durkheim posited a later theory of criminology that had a more deterministic than Lombroso. He believed people were the products of society, rather than the other way around, and felt that crime was a natural and even beneficial necessity (he believed that it help spur greater reforms in law and social justice). His countryman Alexandre Lacassagne helped bridge the gap between the "nature/nurture" debate by citing both hereditary and environmental factors in creating criminals.
The Chicago School was catalyzed by Clifford Shaw and Henry McKay in the 1940s. They drew upon the concentric zone model of urban growth described by Robert E. Park and Ernest W. Burgess some 15 years earlier. It stated that crime remained worse in "high transition" parts of the city, where people move often and poverty facilitated breakdown of standard social contracts. As neighborhoods go downhill, the more affluent depart for safer areas, which increases the rate of descent and fosters a culture of crime. This social ecological model of criminology continues to hold a great deal of influence today.
Strain theory arose from sociologist Robert K. Merton, who worked at about the same time the Chicago School was fermenting. He noted the dichotomy between the "American Dream" of prosperity through hard work and the inequities people actually experienced in real life. The resulting strain leads some to resort to crime in order to bridge the gap, as well as joining rebellious social groups, such as street gangs or vagrants.
Modern criminology incorporates different aspects of these theories as well as numerous smaller ones. The primary purpose is to identify the causes of crime and thus to seek amenable solutions (either through crime prevention, punishment rehabilitation, or alleviation of the social stresses which lead to crime). Most criminologists don't believe that crime will be erased in our lifetime or even many lifetimes (those who follow Durkheim's thinking may claim that society would suffer without at least a certain amount of crime). But by better understanding it as a social phenomenon, they can help reduce its negative impact, while helping law enforcement pursue policies which make society a better place for all.