A bathroom sink's stopper mechanism does not look very complex from the topside of a sink, but in reality there are several more parts to the mechanism that you will not see unless you look at the underside of the sink. Broken or loose parts, as well as a maladjusted mechanism will affect the stopper's ability to raise up.
Lift Rod and Clevis Strap
The lift rod is the part of the sink's stopper that you use to raise and lower the stopper in the drain opening. If you look under the sink, which you may need a flashlight to see clearly, the end of the lift rod goes through a piece called the clevis strap. The "C" shaped end of the clevis strap has two holes the lift rod goes through, as well as a small screw that you tighten to keep the lift rod from slipping out of the holes in the clevis strap. The lift rod may have come out of the clevis strap when the screw became too loose.
Check over the pivot rod, or the piece that goes through one of the holes in the other end of the clevis strap and feeds into the side of the sink's drain pipe. Damage to the pivot rod will result in the stopper not raising and will require you to install a new pivot rod. Try moving the lift rod on the top side of the sink while you watch underneath to see how the pivot rod moves. Removing the pivot rod, if necessary, will take you unscrewing the retaining nut off the side of the drain pipe before you can pull out the end of the pivot rod from the pipe.
You must remove the pivot rod from the sink's drain pipe before you can remove the stopper itself. The end of the pivot rod links into a hole on the end of the stopper, pushing the stopper up to raise it. Once you have removed the pivot rod from the pipe, you can pull the stopper out and examine it for damage. If you find damage where the pivot rod feeds into the stopper, replace the stopper and put the assembly back together.
The fact that that stopper will not raise out of the sink's drain opening may simply be a problem of the stopper's assembly being out of adjustment. You can adjust how the stopper moves by loosening the screw on the clevis strap and moving the lift rod up or down before tightening the screw again. You can also change what hole the end of the pivot rod feeds into on the clevis strap. You may need to play with several adjustments until you get the stopper raising out of the drain and also lowering all the way back inside the drain.