The Collaborative International Dictionary of English v.0.48:

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]

WordNet (r) 3.0 (2006):

errand
    n 1: a short trip that is taken in the performance of a
         necessary task or mission

Moby Thesaurus II by Grady Ward, 1.0:

68 Moby Thesaurus words for "errand":
   agency, agentship, assignment, authority, authorization, brevet,
   busywork, care, chare, charge, chore, commission, commissioning,
   commitment, consignment, cure, delegated authority, delegation,
   deputation, devolution, devolvement, duty, embassy, empowerment,
   entrusting, entrustment, executorship, exequatur, exercise,
   factorship, fish to fry, full power, homework, job, job of work,
   journey, jurisdiction, labor, legation, license, lieutenancy,
   make-work, mandate, matters in hand, mission, odd job, office,
   piece of work, plenipotentiary power, power of attorney,
   power to act, procuration, project, proxy, purview, regency,
   regentship, responsibility, service, stint, task, things to do,
   trip, trust, trusteeship, vicarious authority, warrant, work