Behavior alerts are a currently disabled feature of the Player Behavior Systems. Behavior alerts inform you with pop up window in the client if you have recently received an unusually high amount of reports.
The purpose of these alerts was to prevent players from tilting and encourage them to relax when the system noticed they have shown signs of frustration/aggression the last few games. This was supposed to prevent players from becoming toxic in their next few games.
The feature also had systems in place that prevented reports from report abusers or toxic players from having any impact .
The appearance of the behavior alerts was based on getting several reports in multiple recent games, not getting multiple reports in just one game .
Behavior alerts seem to be deactivated since the beginning of 2016. The exact date and reasons are unknown. The fact that the FAQ about behavior alerts on the Riot Support Page is not available anymore  indicates that this feature was removed intentionally.
Update: By now we have confirmation that the removal was intentional, due to the fact that Behavior Alerts confused innocent players and lead to a lot of Support tickets