The tarot inherited its four suits from the ordinary playing cards that reached Europe in the later fourteenth century. The trumps came afterwards, in fifteenth-century Italy, and the divinatory reading later still.
An unchanged suit
A surviving Mamluk pack in the Topkapi Palace collection in Istanbul carries cups, coins, swords and polo-sticks. Swords passed into the European Latin suits without alteration. Italian packs call the suit spade, Spanish packs espadas, and the word means swords in both.
How the English deck got the word spades
The French suit signs devised in the later fifteenth century replaced the Latin objects with simple printable shapes, and the sword became the pike, pique in French. When those cards reached England the sign kept a name borrowed from the Italian suit it had displaced, spade, which is the Italian for swords and has nothing to do with the digging tool it now looks like. The suit of clubs acquired its English name by the same route, from Italian bastoni.
Air, or fire
The elemental attribution of this suit is contested in the same way as that of Wands. The Golden Dawn system assigns Swords to Air and Wands to Fire, which most modern decks follow. Other writers reverse the pair. Neither arrangement is older than the eighteenth century, and the suits arrived in Europe with no element attached to any of them.
Sources The Game of Tarot: From Ferrara to Salt Lake CityMichael Dummett with A Wicked Pack of Cards: The Origins of the Occult TarotRonald Decker, Thierry Depaulis and Michael Dummett