For the usual "hair clog" affecting that type of sink, there's also a long narrow flexible barbed tool that can be shoved down the drain to pull things that snag on it back up for disposal in trash.
Look under the sink, if applying a plunger to the top fails you.
You can either disconnect the linkage so the stopper can be removed, or (in most cases) remove the trap (sometimes not possible, if an annoying plumber has done a glued-in-place trap to ensure future business by making the trap difficult to service.)
Many of those stoppers can be installed to be removable (typically by twisting 1/4 turn) or not removable by where the drain-rod is inserted on the bottom part. If yours is one of those, you'd only have to disassemble the drain rod to get it out once, if you put it in the other part when reassembling.
The side that's up is removable by twisting counter-clock-wise. The side that's down is removable only by removing the drain rod. Pardon the appearance, it must be 50 years old. Again, annoying plumbers like to put them in the non-easily-removable way. Though sometimes there's a valid case for limiting the size of what children can easily drop in the drain, if they are the sort of children that learn to pull the stopper out...