A long time ago, my mom taught me two inexpensive ways to polish silver.
The first involves rubbing an ash and water paste on the silver until it becomes clean. It's definitely cheaper than buying cleaner from the store, but it takes a lot of effort... as well as a handy stockpile of ashes, which I don't have.
The second method is much more entertaining. Simply add baking soda, a dash of salt, and a folded sheet of tin foil to some boiling water.
Then place your jewelry in the water and stir it up.
All done! Here's a video showing what this method does to a tarnished silver spoon:
So what's the chemistry behind it all? Basically, the baking soda and salt provide ways for the tarnish to move off of the silver and onto the aluminum. The baking soda and salt are just the vehicles that allow that process to happen. The silver tarnish (silver sulfide, or Ag(2)S) gets stripped by the hot baking soda solution, then it gets placed onto the aluminum. Here's the chemical reaction:
3 Ag(2)S + 2 Al --> 6 Ag + Al(2)S(3)
It's interesting that this does not occur without the aluminum. One might think the silver sulfide could just dissolve in the water and that would be that. But the only reason it leaves the silver metal is because the electron affinity of the aluminum is stronger than that of the silver. In other words, aluminum is a sexier place to be for sulfur elections. Sorry silver, aluminum just has a better affinity for electrons.
Wait, what about the salt (NaCl)? The salt makes the process happen faster... a LOT faster (think 15 seconds instead of 6 minutes). A quick Google search on polishing silver will mention the aluminum foil/baking soda method, but none of them mention the salt. That's the part that makes it entertaining ;)