The English name of chess is really the name of one move. Medieval Europeans named the game after the warning called out when a king came under attack.
The game itself is believed to have begun in India. Its early form, chaturanga, flourished there by the 6th century. The Sanskrit name literally means "having four limbs," usually read as the four divisions of an army: infantry, cavalry, elephants and chariots.
Around 600 CE the game reached Persia as chatrang. After the Arab conquest it became shatranj, reshaped because Arabic lacked the "ch" and "ng" sounds. Muslim players carried it to Spain and Sicily by the 10th century. Spanish still keeps that older name in ajedrez.
Most of Europe went another way. Persian players called out "Shāh!", meaning "king," when they attacked the opposing king. Many European languages built the game's name from that word, such as Italian scacchi and German Schach. In Old French the warning was eschec, and the game was esches, literally "checks." English borrowed it in the 13th century, and esches became chess.
The final cry gave us checkmate, from Old French eschec mat and ultimately shāh māt. Its meaning is less settled than it looks. The Arabic phrase is usually glossed "the king died." Etymonline cites Barnhart's argument that the Persian original meant something closer to "the king is helpless," or simply stumped. Russian names the whole game shakhmaty, "checkmates."
The same shah turns up in government. Medieval Latin called a chessboard scaccarium, which became Anglo-French escheker. Under the Norman kings, the royal department that handled revenue took that name. Its officials reckoned accounts with counters on a checkered cloth that looked like a chessboard. That is the exchequer. The x in its spelling comes from a mistaken belief that the word was a Latin compound with ex-.
Etymonline also suggests the banking sense of check may have been influenced by exchequer. If so, a Persian word for king runs through the board game, the treasury and the slip you write to pay a bill.