If your shower drain gets blocked with hair or soap, which is not an unusual problem, you can't effectively clean it while the drain cover is in place. Fortunately, the drain cover is usually easy to remove. Yours may or may not have screws holding it down.
Removing the Screws
Look for screws on your drain cover. If you see any, they will probably be No. 2 Phillips screws, and you shouldn't have any trouble removing them with a screwdriver. If one is stuck, spray plenty of lubricant or penetrating catalyst before trying to force it to avoid stripping the head. Once you remove the screws, set them on the floor outside the shower to prevent losing them down the drain.
Prying the Cover
After removing any screws, you can usually pry the cover by hooking a screwdriver through one of the grid openings near the rim and pulling down on the handle. It should pop out, and if there is any caulk underneath it, you can clean that with putty knife. If the cover doesn't come up easily, loosen it by tapping around the rim with a flat-head screwdriver and hammer. Once it's loose, use the screwdriver to pry it away from the drain.