Luncheon plates are sized somewhere between a regular dinner plate and a salad plate -- typically around 8.75 - 9.5", although that can vary depending on the manufacturer. The name is an obvious reveal to their purpose: to be used at a luncheon where presumably one is eating smaller portions or more classic lunch foods such as a sandwich.
I bought my set of luncheon plates for a number of reasons, mostly because my new kitchen is very (very) small and has limited storage space. So their smaller size and multi-purpose applications (lunch plate, dinner plate, salad plate) is very practical. But its also true that I don't eat large dinners so much these days, so the size suits me perfectly.
It's likely that I would not use my new plates in a more formal dinner party situation but my tiny kitchen comes with a tiny apartment, so I won't be in that position for a while. Luncheon plates are a bit harder to find these days as we continue to move away from more formal place settings, but they're out there. Sometimes a salad plate can stand in, but I find the extra inch or so that a luncheon plate offers is just perfect.
Rough guideline for plate sizes:
Dinner plate: 10" - 10 3/4"
Luncheon plate 9" - 9 1/2"
Salad plate 8" - 8 3/4"
Bread and butter plate 5" - 7 3/4"
(Image: Spode Blue Italian Luncheon Plate)