A Successful Attack is simply an attack which does not miss. (Several things can cause an attack to miss, such as the attacker being Blinded, the attack being Blocked, or other abilities such as Horrific Aura.) Thus an unsuccessful attack is one which does miss.
An attack does not need to do damage to be successful. A champion may be Impervious, immune to the damage type, or able to Deflect the damage, however it will still be a successful attack if it doesn't miss.
A listing of effects which trigger on successful attacks can be found here