Hair on the stopper
Lift-and-turn, push-pull and toe-touch stoppers all catch hair below the cap.
Water stays in the tub long after a bath or shower, or barely moves at all.
Short answer
Hair caught on the stopper or in the trip-lever linkage is the most common cause. If cleaning those does not help, the trap under the tub is clogged. If the toilet gurgles or water rises in the tub when it flushes, the main line is blocked, which is a different problem.

Lift-and-turn, push-pull and toe-touch stoppers all catch hair below the cap.
The plunger or spring behind the overflow plate collects hair.
Hair and soap form a plug in the trap under the tub.
Water backs up into the tub because it is the lowest fixture on the line.
Tub drains in many Miami homes run in the slab, so a plumber usually clears them through the overflow opening with a drum auger. If water comes up in the tub from another fixture, the problem is the main line, not the tub. The tub drain page explains the clearing process. If the tub drains slowly only when the stopper is installed, the stopper itself may be set too low. Adjusting it can solve the problem without clearing the drain. Lift-and-turn and toe-touch stoppers both have a height adjustment on the post below the cap.
Call when the tub stays slow after cleaning the stopper and linkage, it fills when the toilet flushes, or there is a sewer smell.
Hair on the stopper or in the linkage, or a clog in the trap below the tub.
Most unscrew counterclockwise or lift and turn. A photo helps identify the type.
Yes. Cover the overflow with a wet cloth so the plunger can build pressure.
The main line is blocked. Stop using water and call.
They often fail on hair and can damage finishes and older pipes. Avoid them.
A partial clog shows up when a lot of water drains at once.
Usually not, unless other fixtures are slow too.
Use a hair catcher, clean the stopper monthly and rinse sand off outside after beach days.
The clog is in the tub's own trap or drain, not the shared line.
Most unscrew counterclockwise when open. A screwdriver may be needed for the post below.