In general, state governments are responsible for protecting their citizens from crime and prosecuting criminals who commit crimes within their state. States are responsible for creating criminal laws and carrying out those laws. The vast majority of criminal prosecutions are conducted by individual states.
However, there are many areas of criminal law in which federal authorities are increasingly becoming involved in. In the past several decades, Congress has enacted a large number of drug statutes. The rationale for such statutes is that drug crimes need a uniform response nationwide, and that it can be difficult to prosecute drug crimes on a state by state basis.
There are also many federal gun laws and federal computer laws. The computer laws are designed to make it possible to punish crime that occurs on the Internet, such as child pornography. Federal gun laws are designed to provide uniformity across state lines for gun offenses.
Some other common types of federal crimes are anything involving federal income taxes or white collar crime. Also, federal authorities are responsible for prosecuting any crimes that occur on federally owned land, such as a national park, even though that land is located inside a particular state.