You need to determine 2 things: did you get paid in 2015? and what type of employment do you have?
Did you get paid in 2015?
You stated in the comments that your first paycheck from this employer was dated January 5th 2016. So the answer is no you didn't get paid in 2015. For most taxpayers the date of the check determines the year it falls under.
That means for you and this employer, your income falls under 2016. That means that you will settle your taxes for this income in April 2017. Your employer will report this income via a 1099 or W-2 in January 2017.
what type of employment do you have?
In some situations it is hard to tell what type of employment you are under. It is important to determine your situation.
If you are getting benefits such has health insurance, vacation, sick leave, or holiday pay then you are an employee.
Did you fill out a Federal W-4, and depending on your state there could have been a state version of the W-4? The W-4 would have asked your marital status, and the number of exemptions you were claiming. The W-4 is to tell Human Resources (HR) or the accounting department to withhold federal and state income taxes. If you did fill one out you are an employee.
If your aren't receiving benefits, and there was no W-4 completed then they are probably treating you as a contractor.
W-2 or 1099?
If you are not an employee then you are considered a contractor. In either case your income will be reported to the IRS. Employees via the W-2, contractors via the 1099 in January of the next year. You will be sent a copy for your tax records.
In many states your employer is supposed to provide you with a pay stub. In many cases this stub is sent electronically to an email address you provide, or it is available on a third party site. Check with your company HR to see where the stub is. The presence of the stub and withholding for taxes and social security will mean that you can be sure there will be a W-2 in January 2017.
The issue for you is that if you are a contractor they will not be withholding taxes for the IRS, or the state taxing authority; they will also not be withholding for social security.
If you wait to determine your status then in the spring of 2017 your taxes could get very complex and expensive. Once you determine the situation consult a tax adviser for how to proceed.