In spite of your best efforts your jewelry has tarnished anyway.
Well, don't despair. Tarnishing is a natural reaction and pretty much impossible to stop altogether, so it's a good thing there are a number of ways to remove it. One important thing to keep in mind is to work on getting rid of it ASAP, before it gets too hard to remove, or even causes pitting and damage to the surface of the jewelry.
First things first -- sometimes silver jewelry is tarnished on purpose -- chemically oxidized -- to bring out pattern. Be sure, if this is the case with your jewelry, that you remove any tarnish carefully enough that you leave the oxidation on. Try going a very conservative route and wash the jewelry with warm water and a bit of dishwashing soap.
One popular way to remove tarnish is to use a silver dip. These can be found in most drug stores, usually near the nail polish. They are simple to use and very effective. You just dip the jewelry, usually for only a matter of seconds, then rinse and you're good to go. I do keep this in my studio, but it isn't my preferred method to remove tarnish. I have found that jewelry treated in this fashion tends to tarnish again much more rapidly. So, be sparing with these. They are quite useful though, if the piece is intricate and there are lots of hidey holes for the tarnish. This is not a recommended method if your jewelry has been chemically oxidized.Also best avoided if your jewelry has stones in place.
Another method is the tried and true silver polishing cloth. There are a number of these available and you can probably pick one up at your friendly neighborhood jeweler's. I use Sunshine cloths. They would fall under "abrasive" methods if we had to categorize them, but they are very mildly abrasive. These are good to use if you have jewelry that has been oxidized to bring out the pattern, as you can use the cloth carefully just to hit the silver highlights and not to get into the recesses of the pattern. Also, you can obviously work around any stones that are in place.
If you don't have any silver dip or Sunshine cloths, I bet you have the ingredients for an effective tarnish remover right in your pantry -- aluminum foil, baking soda and water. Use a disposable aluminum pan or line a baking dish with aluminum foil. Add baking soda and then boiling, or at least very hot, water. Dip your pieces in this and watch as the aluminum foil attracts the tarnish away. You may have to dip a piece more than once, or leave it in for a bit, but this method works .Again, not best to use with stones in place.
If you are a person who makes your own jewelry and are fortunate enough to own a tumbler, this is a great way to remove tarnish. I often just toss my dirty jewelry into my tumbler for a half hour or so, and it comes out clean and incredibly shiny. Some stones can even go into a tumbler, depending on their Mohs hardness value, but that's for another time.
Don't use toothpaste or dry baking soda as a scrub, as they are too abrasive for most jewelry. If you have jewelry with a matte finish, you might try a softened piece of 600 grit Scotchbrite pad, which can be found in the furniture finishing section of your local hardware store.
If all else fails, you may have to take the piece to a professional jeweler for cleaning.