How do I remove the sink drain in this situation?
Here's a diagram of the drain:



It looks to me like you've unscrewed the nut, but not the flange (which is a separate piece that screws to the inside of the tailpiece, while the nut screws to the outside of the tailpiece). 

If things are not loosening easily, you can try some penetrating oil or WD-40 and letting it sit for a while, it may help if there is any rust or corrosion on the threads. The place this would cause a problem if anywhere would be on the outside at the top of the tailpipe.