board (n.1) Look up board at Dictionary.com
Old English bord "a plank, flat surface," from Proto-Germanic *burdam (cognates: Old Norse borð "plank," Dutch bord "board," Gothic fotu-baurd "foot-stool," German Brett "plank"), from PIE *bhrdh- "board," from root *bherdh- "to cut." See also board (n.2), with which this is so confused as practically to form one word (if indeed they were not the same word all along).

A board is thinner than a plank, and generally less than 2.5 inches thick. The transferred meaning "food" (late 14c.) is an extension of the late Old English sense of "table" (compare boarder, boarding); hence, also, above board "honest, open" (1610s). A further extension is to "table where council is held" (1570s), then transferred to "leadership council, council (that meets at a table)," 1610s.
board (n.2) Look up board at Dictionary.com
"side of ship," Old English bord "border, rim, ship's side," from Proto-Germanic *bordaz (cognates: Old Saxon bord, Dutch boord, German Bord, Old High German bart, Old Norse barð), perhaps from the same source as board (n.1), but not all sources accept this. Connected to border; see also starboard.

If not etymologically related to board (n.1), the two forms represented in English by these words were nonetheless confused at an early date in most Germanic languages, a situation made worse in English because this Germanic root also was adopted as Medieval Latin bordus (source of Italian and Spanish bordo). It also entered Old French as bort "beam, board, plank; side of a ship" (12c., Modern French bord), either from Medieval Latin or Frankish, and from thence it came over with the Normans to mingle with its native cousins. By now the senses are inextricably tangled. Some etymology dictionaries treat them as having been the same word all along.
board (v.) Look up board at Dictionary.com
verb senses derived from various senses of board (n.1) and board (n.2) include "come alongside" (a ship), mid-15c. (from n.2); "put boards on, frame with boards," late 14c. (implied in boarded, from n.1); " to get onto" (a ship), 1590s, transferred from mid-19c. to stages, railway cars, aircraft, etc. (from n.2). Meaning "to be supplied with food and lodging" is from 1550s (from n.1 in transferred sense). Transitive meaning "provide with daily meals and lodging" is from 1590s. Related: Boarded; boarding.