What are the different classifications of stainless flatware, how do you tell the difference and how do you care for them?

What do the terms 18/10 Stainless Steel and 18/8 Stainless Steel mean?
The "18" refers to the amount of chromium in the in the steel, in this case 18%. Chromium gives the steel it's stainless properties. The "10" refers to the amount of nickel in the alloy, in this case 10%. Nickel gives the steel greater protection against corrosion and a softer shine.

Stainless steel manufactures state that there is practically no difference between 18/10 and 18/8 stainless, the categories having been created mainly by marketing departments to sell their wares. It is generally accepted that both these designations have the same practical qualities in cutlery of maximum stain, corrosion resistance and "dishwasher safeness".

What about 18/0 Stainless Steel?
18/0 means that there is 18% chromium and 0% nickel. This grade of stainless steel is less expensive to manufacture and therefore costs less to purchase. Though it is not as corrosion resistant as the other grades, is usually considered dishwasher safe and has been used daily for many years in American households.

How Can You Tell If It's 18/10 Or 18/0?
Check it with a magnet. If it sticks to a magnet it's 18/0, if it doesn't it's 18/10. Knives can be a little different since the blades are often made from a different alloy to allow them to maintain a sharp edge longer. Even the blades of the highest quality 18/10 flatware are often magnetic.

Use and Care of Your Stainless Steel Flatware
While almost immune from stains and corrosion your stainless flatware can not totally neglected and expected to give you maximum durability and longevity. Discoloration can be caused by ingredients found in tea, coffee, salad dressing, vinegar, and salt.   Pitting or spotting can be caused by hard water and foods with a high salt content (such as butter). Don't let foods sit caked on your flatware for long periods of time or let your flatware soak in water for long periods of time or overnight.

(Some manufacturers also recommend avoiding dishwashing detergents that contain lemon.)

They named it stain-less after all!
Remember that they named it stain-less, not stain-proof -- corrosion resistant, not corrosion proof. If stubborn stains or rust occurs, use a non-abrasive stainless or metal polish paste or cream to clean the item.

Personal Experience
Though I can not guarantee that you'll have the same results I can tell you that I've used the same set of Oneida 18/10 Flatware since 1968 and it has years of quality service left in it. I load the dishwasher after using the flatware, almost always without rinsing anything, and run the dishwasher once a day, sometimes every two days. That is not what's always recommended, but it has worked for me.