'To run an errand' definitions:

Definition of 'To run an errand'

From: GCIDE

Errand \Er"rand\, n. [OE. erende, erande, message, business, AS. [ae]rende, [ae]rend; akin to OS. arundi, OHG. arunti, Icel. eyrendi, ["o]rendi, erendi, Sw. [aum]rende, Dan. [ae]rende; perh. akin to AS. earu swift, Icel. ["o]rr, and to L. oriri to rise, E. orient.] A special business intrusted to a messenger; something to be told or done by one sent somewhere for the purpose; often, a verbal message; a commission; as, the servant was sent on an errand; to do an errand. Also, one's purpose in going anywhere. [1913 Webster]
I have a secret errand to thee, O king. --Judg. iii. 19. [1913 Webster]
I will not eat till I have told mine errand. --Gen. xxiv. 33. [1913 Webster]
2. Any specific task, usually of a routine nature, requiring some form of travel, usually locally. An errand is often on behalf of someone else, but sometimes for one's own purposes. [PJC]
3. A mission. [PJC]
To run an errand, To perform an errand[2]. [PJC]