General:
* Chore: A routine task, especially a household one.
* Task: A specific piece of work to be done.
* Mission: A specific assignment or purpose.
* Assignment: A task given to someone.
* Duty: A task that one is morally or legally bound to perform.
Focusing on the purpose:
* Commission: A request or order to do something.
* Favor: A kind act done for someone else.
* Service: An act of help or assistance.
Informal:
* Quick trip: A short journey made for a specific purpose.
* Run: A short trip, especially one made to a nearby place.
* Pick-up: A trip made to collect something.
* Drop-off: A trip made to deliver something.
The best synonym for "errand" depends on the context. For example, "chore" might be used for a routine household task, while "mission" might be used for a more important or complex task.