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1.
When an administrator in online games, message boards, or servers, misuse their power. Often by becoming power hungry and causing trouble just for their own amusement.
Admin Abuse in Counter Strike Source:

Person A (not Hacking): Kills the admin.
Admin: Gets angry.
Admin: Kicks person A from the server for "hacking"
by Sadisticmgt July 01, 2006
 
2.
A phenomenon that occurs on internet message boards where administrators and/or moderators abuse their power to consistently edit annoying, lame or problematic users' posts or profiles to change the context and make the user look like a fool.
Before AA:

Upstanding User A: "How's the new GTA?"
Annoying User B: "God it sucks, it's so lame; if you buy it you're gay."
Upstanding User C: "Don't listen to the above guy, it's great."

After AA:

Upstanding User A: "How's the new GTA?"
Annoying User B: "I hear its amazing when the famous purple stuffed worm in flap-jaw space with the tuning fork does a raw blink on Hari Kiri Rock. I need scissors! 61!"
Upstanding User C: "Don't listen to the above guy, it's great."
by Siegfried Zaga May 21, 2005
 
3.
When someone abuses the power they are given, as an administrator. By unfairly banning or editing posts by regular users on forums etc.
"Omg, who gave Kyle admin! He changed my post to say "HAHAHA DISREGARD THAT I SUCK COCKS" He is the definition of Admin Abuse.
by Apollo13 May 19, 2009
 
4.
When an admin fucks over a player for no ideal reason or just does it for the lulz
Admin Abuse in Counter Strike Source:

Admin: !Slay Noob
Noob: ADMIN ABUSE
Admin: !ban Noob 9999999
Noob2: Lulz
Admin: !rape Noob2 anal
by Quakestriker May 17, 2010