This is a difficult question, but here are some things to consider in order to keep a closer eye on what your child is doing online.
Keep the computer in a common area, like the kitchen. It is easier to see where your child is online when you can actually see the screen. If you aren't sure where they are, ask them.
Check history. You should be able to use the browser's history to go back and see where they have been. However, as the child gets older they will probably learn ways to delete that information.
Use parent controls. You can set up different accounts that will allow each user to have different permissions. So depending on how the person logs on, the computer will allow him/her access to certain places on the Internet.
Get software. There is software out there that you can purchase that restricts users from going to certain sites. The nice thing about this option is that the firewalls are updated often. (Firewalls are the protection between the user and the Internet - they filter to sites that are not appropriate for a particular age group.)
Know your child's username(s) and password(s). Tell them that you need that information if they want to use the computer. Then go in and check - often. this would include email.
Don't give out your password and change it often. Kids are smart and will try to find an excuse to get your username and password sot they can log on as you in order to go anywhere on the Internet.
Friend them on Facebook or other social networking site. Go in often to see what is written on their wall or other things they are posting, including photos. While you are in there, see what their friends are saying. (That can be an eye-opener.)
If you're not tech savvy, learn. Many public libraries offer free computer classes. You need to understand what is out there and how to get to it yourself. Face it, today's kids are much smarter about the Internet than we are.
Why not download some software that can help you monitor your child's online activity? You can try the Time Doctor, you can view whatever activities he's doing and he can also share his screen with you using another computer.