The most common problem that crops up with a pop-up sink stopper in a bathroom sink is simply a build-up of hair and particulates that clogs up the sink mechanism and keeps it from working. Other problems can occur when the clevis--the long rectangular bit of metal that links the drain lift rod with the pop-up stopper mechanism--gets out of alignment, as this will cause the pop-up stopper to be unable to fully rise or lower in the sink. Fortunately, both of these problems are relatively simple to fix.
Twist and lift the pop-up stopper to try and disengage it from the lift rod. If the stopper does not come out of the sink as you twist and lift, it is likely hooked onto the pivot rod and must be disconnected before you can pull it out of the sink.
Move under the sink and search for the large plastic retaining nut on the drainpipe. The pivot rod mechanism will go through the centre of this nut and into the drainpipe.
Unscrew the nut--using the pliers, if necessary--and pull the pivot rod straight backward to disengage it from the pop-up stopper. Lift the pop-up stopper out of the sink.
Clean all the hair and other particulates off the pop-up stopper, using the rag and some water and soap. If the pop-up stopper appears worn, or of the rubber seal is cracked or missing, replace the entire stopper.
Place the stopper back into the drain, doing so in such a way that the hole at the bottom of the stopper is facing out toward the pivot rod assembly.
Push the pivot rod assembly back through the centre of the retaining nut and into the drain pipe so that it hooks up to the hole at the bottom of the stopper. Use your pliers to tighten down the nut securely.
Search for the set screw on the clevis--the clevis is the rectangular piece of metal that connects to the pivot rod--near the top of the clevis.
Loosen the set screw with your flathead screwdriver, then adjust the clevis up or down as needed to ensure that the pop-up stopper opens and closes properly when the lift rod is pulled up or down. Tighten the set screw back down with your screwdriver.