Old English, "move swiftly by using the legs, go on legs more rapidly than walking," also "make haste, hurry; be active, pursue or follow a course," and, of inanimate things, "to move over a course."

The modern verb is a merger of two related Old English words, in both of which the initial two letters sometimes switched places. The first is intransitive rinnan, irnan "to run, flow, run together" (past tense ran, past participle runnen), which is cognate with Middle Dutch runnen, Old Saxon, Old High German, Gothic rinnan, German rinnen "to flow, run."

The second is Old English transitive weak verb ærnan, earnan "ride, run to, reach, gain by running" (probably a metathesis of *rennan), from Proto-Germanic *rannjanan, causative of the root *ren- "to run." This is cognate with Old Saxon renian, Old High German rennen, German rennen, Gothic rannjan.

Watkins says both are from PIE *ri-ne-a-, nasalized form of root *rei- "to run, flow," but Boutkan's sources find this derivation doubtful based on the poor attestation of supposed related forms, and he lists it as of "No certain IE etymology."

Of streams, etc., "to flow," from late Old English. From c. 1200 as "take flight, retreat hurriedly or secretly." Phrase run for it "take flight" is attested from 1640s.

Also from c. 1200 as "compete in a race." Extended to "strive for any ends," especially "enter a contest for office or honors, stand as a candidate in an election" (1826, American English).

Of any sort of hurried travel, c. 1300. From early 13c. as "have a certain direction or course." By c. 1300 as "keep going, extend through a period of time, remain in existence." Specifically of theater plays by 1808. Of conveyances, stage lines, etc., "perform a regular passage from place to place" by 1817.

Of machinery or mechanical devices, "go through normal or allotted movements or operation," 1560s. Of colors, "to spread in a fabric when exposed to moisture," 1771. Of movie film, "pass between spools," hence "be shown," by 1931.

The meaning "carry on" (a business, etc.) is by 1861, American English; hence extended senses of "look after, manage." As "publish or print in a newspaper or magazine," by 1884. 

Many senses are via the notion of "pass into or out of a certain state." To run dry "cease to yield water or milk" (1630s). In commerce, "be of a specified price, size, etc.," by 1762. To run low "be nearly exhausted" is by 1712; to run short "exhaust one's supply" is from 1752; to run out of in the same sense is from 1713. To run on "keep on, continue without pause or change" is from 1590s.

The transitive sense of "cause to run" was in Old English. By late 15c. as "to pierce, stab," hence 1520s as "thrust through or into something." The meaning "enter (a horse) in a race" is from 1750. The sense of "cause a mechanical device to keep moving or working" is by 1817.

Many figurative uses are from horseracing or hunting (such as to run (something) into the ground "carry to excess, exhaust by constant pursuit," 1836, American English).

To run across "meet by chance, fall in with" is attested from 1855, American English. To run into in this sense is by 1902. To run around with "consort with" is from 1887.

In reference to fevers by 1918. To run a (red) traffic signal is by 1933. Of tests, experiments, etc., by 1947. Of computers by 1952. Time has been running out since c. 1300. To run in the family is by 1771. The figurative expression run interference (1929) is from U.S. football. To run late is from 1954.

mid-15c. (earlier ren, late 14c.), "a spell of running, the act of running," from run (v.).

The Old English noun ryne/yrn (early Middle English rine) meant "a flowing, a course, a watercourse;" the modern sense of "small stream" is recorded from 1580s, mostly in Northern English dialect and American English. The sense of "a flowing or pouring, as of liquid" is by 1814. In reference to the action of a school of fish moving together, especially upstream or in-shore, by 1820.

From 1804 as "place where anything runs or may run." The meaning "the privilege of going through or over, free access" is from 1755. In. U.S. baseball, "feat of running around the bases without being put out" by 1856; the sense in cricket is from 1746.

Meaning "continuous stretch" (of something) is from 1670s. That of "continuous use, circulation, or observance" (as in run of luck) is by 1714. The general sense of "a continuous series or succession" has yielded many specific meanings, as "three or more playing cards in consecutive order" (1870). In music, "a rapid succession of consecutive tones," by 1835.

The financial meaning "extraordinary series or rush of demands on a bank, etc." is recorded from 1690s. The market sense of "sustained demand for something" is by 1816.

From 1712 as "a spell of sailing between two ports;" hence also "an excursion trip" (1819); "single trip by a railroad train" (1857); the military aircraft attack sense (as in bombing run) is from 1916. Hence also "a regular round in a vehicle" (as in paper run, milk run, etc.).

In printing, the meaning "total number of copies done in a single period of press-work" is from 1909. In publishing, "set or series of consecutive numbers of a periodical," by 1889.

Meaning "tear in a knitted garment or stocking" is from 1922, probably on the notion of "a failure caused by looseness, weakness, or giving way;" to run had a specialized sense in reference to machinery, "to slip, go awry" (1846), and in reference to lace it meant "to unravel, come undone" (1878). Also compare running stitch "loose, open stitch" (1848).

Phrase a run for one's money "satisfaction for trouble taken" is from 1872 in a figurative sense, from horse racing, where it implied real competition (1841).