A dark patina can add textural interest to vintage copper and brass, but tarnished silver just looks neglected and old. Fortunately, this black film doesn't actively harm your silver and removing it isn't difficult. There are several methods ranging from commercial solutions to gentler and greener do-it-yourself alternatives.
What Tarnish Is
Tarnish is a blackening of the silver's surface caused by a reaction between the silver and the sulfur molecules in the air. This alchemy between the two causes a dusky film to form on the surface of the silver and in the little grooves and flourishes of ornate pieces such as flatware. New tarnish is simple to remove, while older discoloration can be a bit more stubborn.
Using Silver Polish
Commercial silver polish is available in both liquid and paste form. Apply a spot of liquid or paste no larger than a dime to a clean, lint-free cloth. Fold the cloth and rub it against itself to spread the silver polish. Massage the silver polish onto the surface of your silver in small circles. Use gentle pressure to apply the polish and then go over the same spot with slightly more pressure to wipe away the polish and tarnish. Keep turning and refolding the cloth as it picks up the tarnish so you're always using a clean bit. Otherwise, you'll just rub the tarnish back onto the piece you're polishing. Reapply silver polish to the cloth in clean areas as needed. Rinse your silver in lukewarm water and dry it thoroughly to remove all traces of the silver polish.
Toothpaste Cleaning Method
This method works the same basic way as the silver polish method. Use a plain, white toothpaste with no extra whitening ingredients. Apply a small amount to a clean, lint-free cloth, such as a shop cloth or an old, clean diaper. Rub the toothpaste onto the silver in small circular motions and then wipe it off. Toothpaste isn't specifically formulated to clean silver, so removing the tarnish may take a bit of effort. Use a clean, soft-bristled toothbrush to remove tarnish from ornate scrollwork or filigree and then gently rinse your silver in lukewarm water. Dry it thoroughly with a soft, absorbent cloth.
Baking Soda Method
Line your sink or a deep basin with aluminum foil. Place your silver on top of the foil, making sure that no pieces overlap each other. Sprinkle the silver with a fine coating of baking soda. Carefully fill the sink or basin with boiling water, making sure that you don't pour it directly onto any of the powdered pieces. You can add a few tablespoons of vinegar if you like, but it isn't necessary. Once the water has completely covered the silver, let it soak for 15 minutes or until the water is cool enough not to burn your hands. Remove the silver pieces, rinse them in lukewarm water and wipe them dry with a soft, absorbent cloth. This is the easiest method for very ornate silver pieces, though you may still need to use a soft, clean toothbrush dipped in a thin paste of baking soda and water to coax any remaining tarnish from little decorative nooks and crannies.