Etymology
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check (n.1)

c. 1300, in chess, "a call noting one's move has placed his opponent's king (or another major piece) in immediate peril," from Old French eschequier "a check at chess" (also "chess board, chess set"), from eschec "the game of chess; chessboard; check; checkmate," from Vulgar Latin *scaccus, from Arabic shah, from Persian shah "king," the principal piece in a chess game (see shah; also compare checkmate (n.)). Also c. 1300 in a generalized sense, "harmful incident or event, hostile environment."

As "an exposure of the king to a direct attack from an opposing piece" early 15c. When his king is in check, a player's choices are severely limited. From that notion come the many extended senses: From the notion of "a sudden stoppage, hindrance, restraint" (1510s) comes that of "act or means of checking or restraining," also "means of detecting or exposing or preventing error; a check against forgery or alteration.

"Hence: "a counter-register as a token of ownership used to check against, and prevent, loss or theft" (as in hat check, etc.), 1812.  Hence also the financial use for "written order for money drawn on a bank, money draft" (1798, often spelled cheque), which was probably influenced by exchequer. Hence also "mark put against names or items on a list indicating they have been verified or otherwise examined" (by 1856).

From its use in chess the word has been widely transferred in French and English. In the sense-extension, the sb. and vb. have acted and reacted on each other, so that it is difficult to trace and exhibit the order in which special senses arose [OED]

Meaning "restaurant bill" is from 1869. Checking account is attested from 1897, American English. Blank check in the figurative sense is attested by 1849 (compare carte blanche). Checks and balances is from 1782, perhaps originally suggesting machinery.

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check (v.1)

late 15c., in chess, "to attack the king; to put (the opponent's king) in check;" earlier (late 14c.) in a figurative sense, "to stop, arrest; block, barricade;" from check (n.1) or Old French eschequier, from the noun in French. A player in chess limits his opponent's ability to move when he places his opponent's king in check.

The other senses seem all to have developed from the chess sense, or from the noun: "To arrest, stop," then "to hold in restraint" (1620s); "to hold up or control" (an assertion, a person, etc.) by comparison with some authority or record (1690s); of baggage, etc., "to hand over in return for a check that serves as a means of identifying" (1846); "to note with a mark as having been examined, etc., mark off from a list" (1928).

Hence, to check off (1839); to check up (1883); to check in or out (in a hotel, of a library book, etc., 1909). To check out (something) "to look at, investigate" is from 1959. Related: Checked; checking.

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check (v.2)

"mark like a chessboard, incise with a pattern of squares or checks," early 15c., from Old French eschequier (v.), from the noun in French (see check (n.1)). Related: Checking.

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check (n.2)

"pattern of squares in alternating colors," c. 1400, short for checker (n.1). As a fabric having such a pattern from 1610s.

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check-list (n.)

also checklist, "systematic list intended for reference, verification, etc.," 1849, American English, from check (v.1) + list (n.1).

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spell-check (v.)
"to use a computer's spell checker application on a document," by 1985, from spell (v.) + check (v.1). The applications themselves date to the late 1970s. Related: Spell-checked; spell-checking.
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check-up (n.)

also checkup, "careful examination," 1921, American English, from the verbal phrase (1889), from check (v.1) + up (adv.), on notion of a checklist of things to be examined. The verbal phrase check up (on) is attested from 1889.

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check-book (n.)

also checkbook, cheque-book, "book containing blank checks on a bank," 1872, from check (n.1) in the financial sense + book (n.).

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cross-check (n.)

1903 in research and accounting, from the verbal phrase, from cross (adv.) + check (v.1). As a verb in hockey, "obstruct by holding one's stick across an opponent," from 1901; as a noun by 1968. Related: Cross-checked; cross-checking.

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