Are people watching what you do on the internet?

If you are connected to a network and that other person has remote access to your computer, then yes. That's why when you go to some sites a Warning dialog box pops up and says "(something something something) It is possible for others to view what you are doing. Do you wish to continue?" If you get that popup then that should plainly answer the question. But yes, people can monitor your "web Surfing."