I am 20 in school wondering what exactly I have to pay taxes on, how does the tax system work and how do I know what I have to pay taxes on?

You pay taxes if you work, and they hold it out of your pay check.
You pay taxes if you buy something at most stores
You pay taxes if you own a home or have a car.

If you have income you can if under certain amouts, fill and if over you have to file an annual income tax.

Taxes go to pay for all the things the city, county and the nation provide for you

You must file your tax return (Form 1040EZ or 1040A or 1040) if your income exceeds the filing requirements. On your inocme you will pay all the taxes just like an adult.
Even if you are not required to file and have a refund, you should file to get the refund.
If someone can claim you dependent, then you can not claim your own exemption.