Sounds like your "employer" is treating you as an independent contractor and they will issue you a 1099-MISC at the end of the year.
I put "employer" in quotes because legally they're not your employer with this set up. Legally you are running a small business and selling your services to them on a contract basis and they are your customer. You are sort of an employee of your own business - this is why they call it self-employment.
The money they pay you is your business revenue, any money you spend in association with providing your services to your customer is business expenses. Your net profit - which you will owe tax on - is your revenue minus expenses.
You will owe both the employer, and employee portion of FICA taxes (these fund social security and medicare programs). Each portion is 7.65%, so you will owe a total of 15.3% of your net profits for this tax. Its referred to as "self employment tax" in the IRS tax publications.
You will also owe ordinary federal income tax. That could be anywhere from 0% to 35% of your net profit depending on your income level, filing status, deductions, etc.
If your state imposes an income tax (most of them do), then plan on 0% to 10% for that as well.
A few cities also impose a local/city income tax, which may add another few percentage points.
So when you add all this up, a good rule of thumb is to allocate at least 25% to 35% of your income toward taxes. If you're making more than about $50k at this position then you should probably pay for a 1 hour consultation with a tax professional to see if you need to be setting aside even more.
Oh, and you also need to file quarterly reports and send payments, you can't just wait until the end of the year to pay. Your tax pro can explain that as well.