Titles of people involved – State cases are brought by prosecutors or district attorneys; federal cases are brought by United States Attorneys. State court trial judges have a range of titles, but federal judges are called district court judges.
Federal magistrate judges are used in federal cases to hear initial matters (such as pre-trial motions), but they do not usually decide cases.
The use of grand juries to charge defendants is not required by all states, but it is a requirement in federal felony cases unless the defendant waives the grand jury indictment.
States and the federal government have laws making certain acts illegal, and each jurisdiction is responsible for setting punishments for committing those crimes. A state may punish a certain crime more harshly than the federal government (or vice versa), but a defendant can be charged and convicted under both systems.