A despatch is an official report, written by a senior commander in the field to pass on information about the progress of military operations. Commanders would include in their despatches the names of those deserving attention to their services. Mentions may be for a specific act of bravery or for a period of outstanding service.
If your name appeared in these lists you were said to have been “Mentioned in Despatches”. The despatches were usually published in The London Gazette, so a mention equated to a public commendation.