The thread alert button allows you to follow a thread so you get a notification when there is a new reply in the thread.
How does this work?
Information about the threads you are following is stored in your browser's memory and never leaves your machine.
Are my thread alerts lost when I close my browser?
The behavior is browser dependent, but in all common modern browsers it should persist between sessions (i.e. if you close your browser and then come back to the site in a new browser session).
How many threads can I follow?
You can follow up to 50 threads at a time. If you try to add more than that limit, the last one on the list (sorted by reply activity) will be automatically unfollowed.