A card may only be used in a particular format if the card is from a set that is legal in that format or has the same name as a card from a set that is legal in that format. Zendikar Expeditions and Masterpiece Series cards may only be played in formats where the card is already legal.
The complete format list is below in this section. Note that it is perfectly legal to use old cards, as long as a version of that card is legal in the format in question. For example, a player may use a Beta Lightning Bolt in Modern, even though Beta isn’t a part of Modern, because Lightning Bolt was printed in a Modern-legal set. Masterpiece cards are a special addition to a set, and not actually part of the set itself. Since they are opened in packs they are legal for limited, however they are only legal in constructed formats where the normal edition of the card is also legal.
Cards banned in a specific format may not be used in decks for that format. Cards restricted in a specific format may only have one copy in a deck, including sideboard.
If a card is banned, it cannot be used. If a card is restricted, it doesn’t follow the “four-of” rule above; rather than having up to four copies in a given seventy-five, a player may have at most one copy. As explained in section 6.5, Vintage is the only format with a restricted list.