Am I considered a self employed dental hygienist? If so, do I just pay income tax, etc. at the end of the year?

Sense they are not taking taxes out of your check, it sounds like you are self employed/contract labor. This being the case the taxes are payed quarterly, the IRS will give you forms and tax tables to use. If there is only 1 job and part time your tax liability is minimum. The kicker is your social security taxes are to be paid even if your federal and state end up being $0. Technically nothing is due until the end of the year, but is due and payable then. Penalties and interest will eat you alive, if not paid.