If your copper cookware, jewelry or artwork is covered in a green substance, then it has been corroded by exposure to air or moisture. While deeply corroded, or tarnished, copper can be difficult to restore to its proper shine, there are a variety of options at your disposal. One or more of them should be just what you need.
One option is to polish the copper with an abrasive. Sandpaper, steel wool and nylon mesh are all effective abrasives, and they are common enough that you may already have one or more of them in your home. You can also try a metal cleaner. Metal cleaners contain silica, which is basically, sand suspended in a gel, giving it abrasive qualities. The rougher an abrasive agent feels, the stronger it is. However, strong abrasives can damage the finish on your copper items, and may leave deep scratches of rough patches. These rough areas will then trap dirt and grease and corrode more easily in the future.
Acids are good at dissolving away copper corrosion. Try wetting a cloth with vinegar or lemon juice and scrubbing the copper until it shines again. Both vinegar and lemon juice are mild acids, and will often be all you need for this job. You can also combine the two approaches. Cut a lemon in half and cover the exposed fruit with salt. Polish the copper with it. The salt is an abrasive and the acid of the lemon juice assists it. Wash the copper with water after using an acid on it, or it will corrode again very rapidly.
If the copper is too badly corroded for simple polishing or a weak acid, then use a stronger acid. Oxalic acid corrosion removers are extremely effective. If you do not have any, then look for a toilet bowl cleaner. Many of them use oxalic acid or another strong acid like sulfuric or hydrochloric acid. Wear latex kitchen gloves and eye goggles when scrubbing with these cleaners, as they can damage your skin and eyes. Throw away any rags or brushes that you used with them after you are finished. Do not try to wash them out, as you could contaminate your sink and eating utensils with them. Clean the copper thoroughly with a wet rag after using strong acids to remove all traces of it.
For really tough problems, soak the copper in a strong acid before trying to scrub the corrosion away. Do not soak it in a tub made out of any other metal than copper. Mixing different metals in an acid can cause chemical reactions that could damage the copper.
As an alternative to acid, try ammonia. Make a mixture of half water and half ammonia, and add enough detergent to make suds. Dip a cloth into this mixture and scrub the copper with it. Wear gloves while scrubbing. Rinse the copper off after scrubbing, and then flush your sink's drain thoroughly with water, to wash all of the ammonia down the pipes before it can damage them.