Currently the Play by Play section of the Match Centre will use this field as shown below. The way this field works for the administrator is if you enter a team nickname this will show, if you don't enter a team nickname it will fall back to the club abbreviation. If neither field is filled in then it will fall back to the team full name.
To enter a club abbreviation edit the details of the relevant club and fill in something in the abbreviation field. For the best display on the website this should be under 5 characters.