Tarnish occurs because surface molecules of the silver react with sulphur from the air, or certain foodstuffs such as eggs, to form a compound called silver sulphide. So when you polish it off, you are actually removing some of the silver. For silver plated items, this means that with repeated polishing over many years there is a danger of wearing completely through the plating and exposing the base metal. The only solution then is to have the item re-plated.
So a better idea is to remove the tarnish by converting the silver sulphide back into silver. This is quite easy to do and doesn't require any sophisticated equipment. You will just need a bowl large enough to allow the silver item to be completely submerged, some hot water, some aluminium foil and some baking soda.
Line the bottom of the bowl with aluminium foil and put in the silver item, making sure it is in contact with the foil. Boil enough water to immerse the item and pour it over. Sprinkle the baking soda into the water, using about 1 tablespoon for each pint of water. It will froth and foam and may spill over the top of the bowl, so best do this in the sink. Straightaway, you should see the tarnish begin to disappear. For lightly tarnished items, it should all be gone in a few minutes. For heavily tarnished items, you may need to re-heat the water when it has started to cool and repeat the treatment.
So how does it work?
Well, it is an electrochemical reaction. In the hot water and baking soda solution a small electric current is generated between the touching aluminium and silver. The electric current causes a chemical reaction between the aluminium and the sulphur because aluminium has a greater affinity with sulphur than silver has. The sulphur in the tarnish is attracted into the solution and towards the aluminium, leaving the silver behind, where it belongs. The reaction happens faster when the solution is hot. The compound formed when aluminium and sulphur react is called aluminium sulphide and that's what you will find floating in the bottom of the bowl or stuck to the foil when you are finished. And your silver will be bright and shiny.