The concept of watched topics is rather simple.
Any topic in which you've posted a reply is considered your watched topic. That includes your own topics.
Additionally, any topic for which you clicked "Watch" link (top left of topic's pages) is also added to the list of your watched topics.
Clicking "Unwatch" link removes the topic from your watched list.
Keeping a list of watched topics allows you to follow up on all topics for which you've shown interest (either by posting there or by clicking "Watch" link).
You can access your watched topics by using Watched Topics link at the top. Additionally, the number in that link indicates how many of your watched topics have new replies.
On Watched Topics page, you can instantly see if there are any new replies in any of your watched topics.
If you select "Show My Topics Only" filter, then you'll only see your own topics. This is the most convenient way to check if you have any new replies.
If e-mail notifications are enabled in your profile, then you'll also receive e-mail whenever a new post is made in one of your watched topics.
Also, on many pages that list topics, there is "=>" link near the topic's title. That link would lead you to the last post you've read in that topic. You can use it to quickly pick up where you left off and read just the new replies, for any topic.
The concept of watched topics is central to the operation of this forum. So please make sure to check the Watched Topics link at the top whenever it's shown in bold and contains a number. That means there are new replies in topics of your interest.