In SO, when your question got answer. or you got new badge, event is triggered. Or when you got new PM in forum, it also lets you know by alerting message.
You see message that something happened with your account when you enter site for first time after this event.
How is this implemented? How do scripts know, that they have something new to show you?
The event happens when another person does something like vote on your answer or question. This is recorded on the server side.
When you log on to the site the fact that some events occured while you were away can be determined from looking up the database for these records.
While you are on the site it is possible for the page to periodically poll the server for changes. So its not really the server that the event is triggered on.
As an answer to your last comment
There are several tecniques to mark events as a 'new'. It can be another field in the database table, of boolean type: telling if event was shown to user or not. Or - easiest one - just a time of last user's visit being recorded in the session, and then al upcoming event's time being compared with it.