What exactly is a 1099 employee?

There is no such thing as a 1099 employee. Either you are an employee (and have taxes withheld; receive benefits; and receive a W-2) or you are an independent contractor receiving a 1099. Note that most people called "inpendent contractors" or "1099 employees" are not--they have been misclassified by their employer and really employees (and need to be paid, etc. as employees).
It doesn't matter what you call them or what forms they fill out: if a person meets the criteria to be an employee, they are an employee. You can find the full criteria on the U.S. Dept. of Labor website or the IRS website, but in brief, if you manage how they do their jobs, if you exercise control over their hours or location, if you provide their tools, equipment, etc., if they can't lose money while working for you (e.g. spend more on their "business" costs then they earn) but are guaranteed to profit (by no matter what, netting positive on pay), etc., they are an employee. They don't have to meet all these criteria to be an employee, but the more of them they do meet, the more likely they are to be an employee. The most important is control--if you exercise control over them, they are an employee.
Example: I used to work in publishing. If we hired a graphic artist to work onsite, during the hours other staff or management was around, and could direct them how to do their job, they were an employee. The same graphic artist, if she worked at home, on her own hours, using her own computer, and was sent a Word file which she turned into a laid-out book for us without a manager looking over her shoulder, would be an independent contractor (1099). 
If you misclassify workers, you could potentially be held liable for all the taxes, benefits, overtime (if applicable), etc. you should have paid them.