The stopper is operated by a lever that is connected to the knob that goes up through the faucet. So find where the lever goes into the back of the drain pipe at the bottom of the sink.
It's held in place by a nut. Loosen the nut and you can pull the lever all the way out. Then you can lift the stopper all the way up and clean it off.
NOTE: this part will be gross. Just so you're warned.
NOTE also: I recommend removing the trap again and running a finger all the way up inside the pipe, or whatever you can to reach up there, and also remove all the goo that is stuck to the side of the pipe, that doesn't come away with the stopper. Otherwise you're leaving behind an accretion of gunk that more gunk will stick to.
When you're done, drop the stopper back into the drain with its lifter hole toward the back of the sink. Re-insert the lever, testing for engagement, before you tighten the nut down. Tighten the nut enough that it holds the stopper in place where you leave it, but then only tight enough to prevent leaks. Don't go crazy.
NOTE: you appear to have a cable operated stopper, but that's only a detail - the connection is threaded and unscrews like any other stopper lever. Instead of pulling out a lever, you'll undo the ferrule, remove the cam and then you can lift out the stopper.
Good luck with it.