1099 contractors are independent business owners (even if they are sole proprietors) who provide services on a contract basis. They tend to have a number of clients, make their own business decisions and hours, and supply their own tools and materials to get the job done. They are fully responsible for paying their own employment taxes.
However, if you work solely for this one employer, work at the employer's premises, and get all of the materials and tools from the employer, then it's more likely that you are an employee and should not be called a 1099 contractor.
If you are an employee, the employer should be withholding and paying in the appropriate federal, state and local taxes on your behalf. The issue is not whether you're "called" a 1099 contractor, the issue is the amount of control the employer has over the how, where, when and what of your work.
You may want to consult with a local employment lawyer to make sure you're getting all you're entitled to.