You can't track a phone for free by its phone number, although there are a number of websites that claim they can do this. I definitely wouldn't bother wasting your time with them - it won't work.
Why should we be able to track a phone by its phone number?
There are only a couple of reasons I can think of for wanting to be able to track someone's phone using their phone number. The first is a definitely moral, i.e. You'd like to know the whereabouts of someone for their own safety, if someone is missing, or to make sure your child is safe. The second is a lot less moral, and involves wanting to know there whereabouts of someone that probably doesn't want you to know where they are. This is a massive violation of someone's privacy, so it's probably a good job we can do it.
That being said, there are ways to track your own mobile phone these days via apps, and these are pretty cool. I personally use the Find My iPhone app that allows me to track my own phone from the internet if it's lost or stolen. I can also wipe the phone completely, so it's useless to anyone that happens to find it.
If you've got an iPhone, this video will help you to set up Find My iPhone:
