It breaks down basically like this:
Cases involving Federal Law: if the court is being asked to interpret the application or violation of a federal law, then the federal court system has jurisdiction. Kidnapping, for example, is a federal crime, so someone accused of that crime can only be charged in federal court.
Cases between different states - States that are in controversy with each other over trade law, contracts, extradition, etc. have their cases heard in federal court. It makes sense that the feds would be the arbiter for disputes between the states. This would also include disputes between citizens of different states.
Cases where the US government is involved - That is, where the Federal Government is the plaintiff or the defendant in a lawsuit.
International disputes - where one of the parties in the case is the US government and another is a foreign government, business or entity. This can include treaty interpretations, or issues with the World Bank or the World Trade Organization.
Cases involving commerce or issues on rivers and oceans - because the territory of rivers and seas are often shared between states or countries, the federal government has jurisdiction over those issues whenever the US, a state, or an American is involved. This is in part because of the Commerce Clause of the Constitution.