Watching a thread means you’ll be notified when someone posts in it. You can choose to receive these notifications by email, on-site alert, or both. You can also watch an entire section, if you so choose.
When on a thread, in the top right corner there will be some links. Next to the Thread Tools drop down, there is a Watch Thread link. When you click it, you will be given a variety of options. Follow the prompts and you will now be watching the thread!