How to take the pop-up stopper out of the sink

Don't bother with all that, just remove the U shaped pipe (The Trap) which should be a pressure fitting that can be easily unscrewed without any tools. Most of the crap will be in that U part of the pipe, the rest you can get out by sticking something like a toilet brush or pipe cleaner up the exposed pipe.

BTW: Be sure to put a bucket under the pipe before you unscrew it, some nasty water will likely leak out.

As for that pivot nut, if you really want to remove it, it USUALLY is easy enough to unscrew with your bare fingers, but if it is rusted you can use a pair of channel lock pliers. If it is just spinning however, it may be all the way off the threads and you just need to pull it back. It is attached to a rod that pivots in the middle of the pipe. You might need to shake it a little to get it loose, it will also probably help to detach the rod that goes from the sink stopper pull to that nut since it will get in the way.

You could check if the remaining piece of pivot rod that you have in hand has magnetic properties (i.e. a magnet will stick to it). If so you can probably be successful in retrieving the broken part from the trap using a tool similar to those pictured below:

Magnetic reach tools can be purchased online, hardware stores or big box outlets. In addition to the type shown above that have the flexible shaft there is another type that has the shaft made out of a soft aluminum wire that can bend slightly as the thing is pushed into the drain line. This latter type is typically a lower cost than the types pictured.