There are two basic types. The crappy plastic ones that usually break in the first year of use, and the sturdier brass ones that can last 5 to 10 years depending on how they are used, how hard your water is and what your tub goes through (if ten people use the tub per day, that will wear it out faster than one person per day). Guess which one you have - lol :-). since you can unscrew the top plug bit, I will assume that what is stuck is the stem. Gently grab it with a pair of pliers and gently pull and wiggle and slightly twist it as the same time - your not trying to pull it out, instead you are attempting too get it to move about and hopefully release the locking mechanism (most likely a spring loaded bearing) to let go. Since it is plastic I can assure you that the spring is made cheaply and most likely has broken, instead of pushing the bearing out when the bearing is depressed, it has lodged behind the bearing becoming a stop. If a little twist and pull and wiggle does not loosen the stem, then you will need to go shopping at a hardware store and figure out what kind of stopper you have. See the name "foot lockstop" and Toe stopper are brand names that have been used to describe the type of drain thus you could have a foot locctop or a toe stopper - or Touch toe stopper, or "stopper and waste plug - toe operated. Due to the fairly large number of different designed "fixing" one requires seeing what is there to know how to proceed. I will assume that the stem can be unscrewed as well. I assume this. Some are pressure fitted and once pulled are not easy at all to put back in. If you can break it out of there you MIGHT be able to replace. Depends on the make and type. So shopping first is important to know what you are dealing with and what your options are to "replace" it. The guy at the store may try too talk you into buying a whole new drain. You really do not want to go down that road - that usually requires taking out dry wall, unless you are lucky and have a crawl space or a cellar below.

I just had the same problem with a brand new (3 months old) vessel sink with a push drain. I was able to get it unstuck by using a small suction cup from the back of a sun catcher. Just dampen a little and suction on to the center of the drain, then lift.

I had one of those and today I took it out take a pair of plyers and a screw driver and a hammer to it. unscrew the top the white piece bust it with the flat head screwdriver and hammer. then when you can take the plyers to the metal thing in the middle turn it with the plyers until it comes up. And then go the local walmart dollar store.... and buy one of those rubber flat things that go over the drain... Problem fixed. Hair in the drain is causing it to stick.