There are a number of methods for removing tarnish from sterling silver. The benefits and drawbacks of each will be discussed below.
Before you take any action, take to heart the advice given to doctors: First, Do No Harm.
It is within your power to reduce a valuable antique to a worthless piece of slag. Countless people over the centuries have destroyed works of art by rough polishing. Please don't join their number. In fact, the word polishing should be equated with sandpapering in your mind... as something to be avoided. Tarnish is not a foreign substance above the surface of the silver. Rather, tarnish is sulphur which has become chemically bonded into the surface of the silver. Scouring away the silver on the surface in order to remove the tarnish is extreme overkill. The only way to remove the sulphur without removing the silver surface is to (gently) reverse the chemistry.
Keep in mind that there are two qualities you are trying to revive: 1) the original mirror surface formed by the silversmith during the final stages of production, and 2) the microscopic scratches which have naturally accumulated over that mirror surface in the course of lovingly using the piece. Abrasion during cleaning will obliterate one or both of these qualities, and thereby most of the value of the piece.
Hand Polishing 
This is the method of care adhered to by serious collectors, as it brings out the beautiful luster of the sterling without destroying its patina. There are a variety of products available for hand polishing, from pastes to creams. It is generally best to stay with well-known silver-care product manufacturers, as some inferior products may cause scratching. (When in doubt, test the product in an inconspicuous area or on a less valuable piece.) Some examples are Twinkle Silver Cream (very gentle), Wright's Silver Cream (slightly more abrasive), and creams and pastes by the Hagerty company. These are readily available through supermarkets, jewelry stores and silver dealers, as well as on the Internet.
The author of this paragraph infinitely favors Hagerty's foam. Not the pink spray made by the same company, but rather the foam. Forget supermarkets. Look for it in department stores which have a wedding gift department. You'll find it in white plastic jars about the size of a baseball. Despite being called foam, it actually looks like chocolate pudding... and smells absolutely nothing like chocolate pudding. Use it as you used to use fingerpaint in kindergarten: with your bare fingertips, rub the foam over the surface of the silver. Keep running your fingertips back and forth until, usually after just a few seconds, you begin to see the spots of tarnish fade. In cases of more stubborn tarnish, you may need to rinse with tap water and again fingerpaint with the foam a second or third time. Don't be impatient and reach for something scratchier to hurry up the process: that's the slippery slope towards destroying a piece.
Some pieces should not be immersed in water. Items containing mirrors, wooden parts, fabric, glue, etc. should be fingerpainted with as little foam as possible, then wiped with a barely damp sponge or cloth. For particularly small areas of silver near parts that must not become wet, a Q-Tip dabbed into the foam will usually do quite well.
Not all yellowness on a piece of silver comes from tarnish. Gold wash or vermeil was often applied to the interior of cups and goblets and on the service end of utensils to protect against corrosive beverages and foods. This wash is usually thin, and can be easily removed with too-abrasive or too-vigorous cleaning. Hagerty's foam will not injure gold wash, but aggressive scrubbing with a cloth can injure it.
Keep in mind that it's not necessary or even desirable to remove every bit of tarnish from a piece of silver. Especially when the piece has fluted or embossed surfaces, it's expected that the deep grooves will acquire a darker hue than the raised areas. The resulting contrast is something to be left intact. Should a particularly stubborn spot of tarnish refuse to fade after, say, half a dozen foam fingerpaintings, it's time to leave well enough alone.
Crucially, avoid injuring any identifying mark which may be present on the piece. American pieces will, if 92.5% silver, have the word STERLING stamped on them, usually on the underside. Older American pieces, at 90% silver (called coin silver), may have nothing more than the maker's mark stamped on them. European pieces, of various percentages silver, will typically have anywhere from one to half a dozen cryptic symbols (lions, anchors, crowns, moons, faces in profile, single or double letters, etc.) hammered into them. It is imperative that these symbols not be worn away: they are the hallmarks by which a piece's age, origin place and maker are indicated. Lose the hallmarks and what had been a verifiable 1754 French handworked masterpiece is now an unidentified could-have-been-mass-produced-last-week piece of junk.
If you are going to use the piece soon after applying the foam, finishing up the wash with gentle dishwashing liquid and lots of cool water is perfectly reasonable. It is also perfectly reasonable to eat from the piece immediately after rinsing just with water. If, however, you are going to put the piece away, let the foam be the last thing you rinse from the piece. A trace of the foam will remain even after you've run enough fresh water over it not to be able to see any more foam. This trace amount of foam will slow down the piece's chemical desire to become tarnished again. Don't put a piece away wet. Gently towel it dry (blotting rather than rubbing) and let it air for perhaps an hour before storing it away.
The remainder of this page is informative but very risky. Remember: you can play one of two roles with this piece... you can either be its protector, or its destroyer. Silver has value, but the scrap value of the metal is often negligible compared to the condition value. If you feel that the radical steps described below must be taken, let it not be you who takes these steps. Instead, sell the piece as-is and get your money out of it before (with all good intentions) you destroy 90% of its value.
For example, a deceptively easy but damaging way to shine your silver is to use tooth powder. Just dust your fingers with a bit of tooth powder and rub over the area you want to shine. You can finish by wiping the piece with a tissue or a piece of cotton - moist or dry. Now go back and look at the surface you've just worked on. It'll have hundreds of visible scratches that were not there before, and its previously warm glow will have turned dull. If you keep doing this, you'll gradually turn the piece's original mirror finish into something resembling aluminum. This is due to the presence of grit in the mixture. Why is this so damaging? Well, for one thing, silver metal is a lot softer than tooth enamel. Tooth powder containing abrasives is the right tool for scouring foreign matter off of the hard surface of teeth, but tarnish is more akin to tobacco stains. The only way to remove a stain with abrasives is to remove surface material, resulting in a damaged surface. On the other hand, gently applied chemistry removes stains while leaving the surface intact.
Wheel Polishing 
In cases of extreme tarnish and/or corrosion, it may be necessary to take the item to a jeweler or silver repair company. There the tarnish can be removed with jeweler's rouge. This is sometimes referred to as putting an item "on the wheel". While this method does remove tarnish, it also removes the patina, or microscopic wear marks which give antique silver its beautiful glow, leaving in its place the shiny look of an item fresh off the factory line. Thus, this method is avoided by serious silver collectors in favor of hand polishing.
See and avoid. In almost every case I've encountered, people who own polishing wheels are obsessed with overusing them... even so called professionals. Jewelers are often called upon to remove a previous owner's initials from, say, a set of silver spoons. I have seen the results of overly enthusiastic jewelers who removed the initials by buffing away a thumb-sized crater, thereby rendering the piece even less valuable than it had been with someone else's initials still attractively inscribed.
Silver Dips 
The term dip calls to mind television infomercials promising instant polishing of your cherished silverware. These commercially sold silver dips are an abomination. A few work by eating away the tarnished surface of the metal, somewhat akin to the idea of removing a tattoo with battery acid. Most dips, however, work by turning the tarnish from its natural dark color to a cloudy white haze, somewhat akin to the idea of trying to hide a tattoo with skin-colored magic marker ink. Either type of commercially sold dip results in a permanently damaged piece. See and avoid.
In the broader sense of the word dip, however, it is possible to try cleaning a piece of silver by immersing it in some sort of liquid as opposed to fingerpainting it with foam. There are several mild chemical combinations which will gently attract some sulphur away from a piece of silver. One of the simplest involves dissolving baking soda in warm water in a glass bowl, then inserting both the silver item and a piece of aluminum foil. You'll know it's beginning to have some effect when you catch a whiff of eggs rising from the water. Frankly, though, I've tried this and several other passive approaches to tarnish removal, and I'm not particularly impressed with the results. Fingerpainting with foam does a far better job of restoring silver, for the same reason that shampooing your hair cleans it better than merely soaking your head in soapy water.
Serious sterling collectors avoid silver dips. One reason is that silver dip tends to brighten areas of the design that should remain dark in order to give a relief effect. This is especially important in pieces that have Repoussé and chasing. Another reason is the difficulty in removing the end product of the conversion, which usually requires an abrasive cleanser... and abrading the surface is one of the most effective ways of destroying a piece's value.
A few cautions should be noted. Sterling silver is a very soft metal that is easily damaged. Be careful not to dent the piece by knocking it against a sink or other hard surface. Be very gentle with sterling hinges, such as those on teapots and boxes, as these can be easily bent by mishandling. Do not use a toothbrush or other bristled brush on sterling, as this will permanently scratch the surface.
A more effective baking soda method: Put several layers of aluminium foil (glossy side up) in a non metallic container, add some baking soda on the foil, place your silver object on top, add more baking soda on top of the silver piece and finally cover all of the silver piece with BOILING water. Wait five minutes.
Frequency of Polishing / Storage 
Sterling that is continuously on display, such as a tea service or candelabra, will need more frequent polishing than pieces that can be wrapped and stored. Much of this depends on the amount of sulphur in the environment (city dwellers living near high automobile traffic, for example). Simple sealable food storage bags, or dry cleaning bags for larger pieces, will inhibit tarnish from accumulating on stored pieces. Make sure that the item is thoroughly dry before storing, however, as trapped water will cause corrosion.
Several sources indicate that storage in plastic bags can do more harm than good, since poor quality plastic emits gases that cause corrosion and tarnish. Make sure you use a good quality polyethylene sealable food bag in conjunction with a silver cloth or acid free paper. This will provide good protection.
Don't wash sterling and stainless together. If they touch, a chemical reaction may occur, resulting in black spots on the sterling. Use a mild dishwashing liquid, and don't apply undiluted dishwashing liquid directly to sterling. Detergents containing phosphate can turn sterling brown.