Plastic drywall anchors have different designs depending on the amount of weight they are designed to hold. Conical plastic anchors are the lightest-duty ones, suitable for hanging pictures, while threaded and winged ones are suitable for heavier loads like lightweight shelving. Conical anchors expand against wallboard when you drive a screw into them while threaded anchors, which resemble large screws, have a slotted head that allows you to use a screwdriver to drive them into the wall. The foldable wings on winged anchors fit through a pre-drilled hole and expand against the back of the drywall to hold the anchor in place.
You can usually coax a conical anchor out of the wall by wedging a screwdriver or utility knife behind the head and prying outward. If the head is buried too deeply or the anchor is wedged too tightly, as it may be if it is installed in a concrete or tile wall, a simple method is to partially drive a screw and use a hammer or nail puller to pry out the screw. The anchor should come with it. Threaded anchors are easier to remove. You can simply unscrew them from the wall with a screwdriver.
The design of winged anchors includes a feature for easy removal. Partially driving a screw collapses the wings enough to allow you to pull it out of the wall. This doesn't always work, however, especially if the anchor has been supporting a heavy load and the wings are wedged against the back of the drywall. When you can't remove a winged anchor in the normal way, an alternative is to cut off the head with a handsaw or knife and push the anchor through the back. It will fall into the wall and hopefully remain out of the way.
Patching the hole left by a wall anchor you removed is straightforward if the wall covering is drywall. You can fill small holes by spreading one or two coats of spackling compound or drywall joint compound. You may have to fill larger holes with patching compound first. If you removed the anchor from concrete, you can usually patch the hole with mortar, but the repair is trickier after you remove an anchor from tile. A good solution is to fill the hole with epoxy putty that is as close to the color of the tile as possible.