Crimes and criminality are classified by several different means. First is the perceived seriousness of the offense. For example a traffic offense of not yielding at an intersection is classified as a misdemeanor. Misdemeanors usually are punishable by a fine or jail time of less than one year. Compare the traffic offense to a different crime like murder or rape. These, of course, are more serious and are classified as felonies. These offenses are punishible by incarceration of more than one year's duration and or a fine. Other examples of classifications are property crimes and crimes against the person. Simple theft is a property crime whereas rape and murder previously mentioned are crimes against the person. Crimes are also classified as violent or nonviolent. Most states recognize the seriousness of violent acts and accordingly have more harsh punishments for violent offenders. Additionally, there is the classification of white collar crime. These offenses are usually nonviolent in nature and are related to such things as fraudulent business transactions and matters involving the conduct of business. There are crimes classified as "victimless" crimes. Prostitution is an example of this type.
Most states in the U.S. compile crime statistics and publish them in U.C.R.'s (uniform crime reports). The UCR's distinguish the types of crimes committed or attempted in a certain geographical region.
There are many different types of crimes and like the first responder suggested there are many ways of breaking them down. I will break them down by type.
Sex Crimes- sex crimes are crimes that revolve around a sex act or attempted sex act. Some examples are: rape, child molestation, child pornography, child prostitution, incest, and etc.
Homicide- is a crime that involves the killing of another person.
Child Abuse- is any kind of abuse aimed at a child. This can be sexual in nature, physical and emotional.
Kidnapping- is the taking of a person without their consent or that of their parent
Money Laundering-is any transaction that which generates an asset or a value as a result of an illegal act.
Below is a link to many other types of crimes.
There are a few different ways I can think of to categorize crimes. I do not know which of them you are thinking of when you ask this question.
One way to categorize crimes is to split them into felonies and misdemeanors. Felonies are the more important crimes and they tend to call for more severe punishments.
Another way to categorize crimes is to separate them into crimes against people and crimes against property.
A third way would be to split them between violent crimes and nonviolent crimes.
I do not know which of these is most helpful to you.