Anyone watching a ticket will automatically receive email notifications when the ticket is updated so you can keep track of what's going on with a particular problem.
The Watch List
The reporter of the ticket along with the assignee (if any) will automatically be added to the Watch List for that ticket. In addition to this, any user with read/write access to the tickets can add themselves or someone else as a watcher by clicking the Watchers tab to see all available users that can be added or removed as a watcher:
From here you can add or remove a user simply by changing the checkbox next to their name appropriately.
If you find an occasion where the user has been pre-checked and the box is grayed out, as the tooltip will explain this is because the user has already set themselves up to always be notified of updates by their own preferences for either the project, or the account as a whole.