State laws cover most crimes, such as murder or theft. State laws can be about almost anything, as long as the law is constitutional. This gives rise to many funny state laws, like making it illegal to sell your own eyeball in Texas. Federal crimes, however, tend to be a lot more specific.
In order to pass a federal law, there must be some reason to do so on a national level. Counterfeiting, for example, is a federal crime. Because the federal government is responsible for printing money, it has jurisdiction to prosecute. Other types of crimes also fall under federal jurisdiction. A crime that involves multiple states or crosses state lines becomes a national issue. Also, any crime that takes place on tribal land or in a national forest is a federal crime.
Sometimes, both the state and federal governments will try a person for the same crime. Double jeopardy prohibits two trials within the same jurisdiction. However, the separate sovereign doctrine permits both the federal government and a state government to bring charges against the accused. For example, per the Texas Penal Code, it is illegal to knowingly possess forged documents, including money. So, both the state of Texas and the federal government could charge a Texas resident who gets caught counterfeiting.
Punishments for federal crimes tend to be more stringent than those handed down by state courts. Federal crimes have severe minimum sentences and tend to be longer than similar convictions at the state level. Those convicted of a federal crime will go to federal prison, rather than to a state institution.