I was an employee at a small business and received a 1099-NEC. The website thinks I'm the employer, which I wasn't, I was the employee. How do I enter this form?

The program will ask who issued the form. In that case, you enter the EIN number and Employer name and address. When it asks if you are doing business under any name other than your own, do not enter anything (unless you have a DBA).  The address is your address (unless you have a separate office location).

You might do well to delete this version and start from scratch.

Here are the steps for deleting unwanted forms in TurboTax Online:

Open your return in TurboTax. (To do this, sign in to TurboTax, and select the Take me to my return button.)

In the left side bar, select Tax Tools> Tools.

In the pop-up window Tool Center, choose Delete a form.

Select Delete next to the form/schedule/worksheet and follow the onscreen instructions

A Form 1099-NEC is used to pay contractors and self-employed individuals, who do not have taxes withheld from their payments, as a W-2 employee does.  
So, TurboTax will ask you to enter in the Self-employment income and expenses section of Wages and Income and will ask questions about your business or self-employed activity. 

This TurboTax Help article explains how Form 1099-NEC works in the return.