the modern verb is a merger of two related Old English words, in both of which the first letters sometimes switched places. The first is intransitive rinnan, irnan "to run, flow, run together" (past tense ran, past participle runnen), cognate with (cf. 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.
Both are from PIE *ri-ne-a-, nasalized form of root *reie- "to flow, run" (see Rhine).
Of streams, etc., from c.1200; of machinery, from 1560s. Meaning "be in charge of" is first attested 1861, originally American English. Meaning "seek office in an election" is from 1826, American English. Phrase run for it "take flight" is attested from 1640s. Many figurative uses are from horseracing or hunting (e.g. to run (something) into the ground, 1836, American English).
To run across "meet" is attested from 1855, American English. To run short "exhaust one's supply" is from 1752; to run out of in the same sense is from 1713. To run around with "consort with" is from 1887. Run away "flee in the face of danger" is from late 14c. To run late is from 1954.