The short answer: 18 refers to 18% chromium content, and the other number refers to the percentage of nickel content. There is no difference between 18/8 and 18/10. It's purely marketing. 18/0 however, has no nickel content, which means it won't keep its silver-like shine over the years.
The long answer:The numbers 18/0, 18/8 and 18/10 refer to the percentages of chromium and nickel in the stainless steel alloy. The "18" refers to the chromium content, which gives flatware its rust-resistance properties, and the "8" or "10" refers to the nickel content, which gives it its silver-like shine and some rust-resistance. There is a lower quality flatware commonly available called 13/0 or 13 Chrome, which simply has 13% chromium content.
These numbers are merely "nicknames" for the lay person to use, and are only used for marketing efforts by flatware manufacturers. When a manufacturer purchases stainless steel from a steel mill, they purchase stainless steel Grade 304, which has a range of 18-20% chromium, and 8-10% nickel content. Grade 304 in flatware is usually at the lower end of that range. To keep the cost down, steel manufacturers will make grade 304 with 8.2% nickel, which clears the legal hurdle of calling it 18/10.
What does all this mean? It means that there is no difference between 18/8 and 18/10 stainless steel in flatware. The difference between the two is purely a marketing effort.
18/0 however, is a different story. It's made from the "Grade 400 series", which are not as corrosion resistant, and the pieces are very magnetic. Lately, more flatware manufacturers have been switching their lower-priced patterns to 18/0 to keep the price down.
