Cup plates are coasters that provide a place to rest a tea cup while leaving space for a light snack. Teacup plates originated in England in the early 1800s and went out of fashion in the second half of the 19th century (Barber puts the peak of popularity in the US at 1840s), with a brief reappearance in the first third of the 20th century as bridge sets. The cup plates were in common use in the United States during the first half of the 19th century, and were a precursor of the very specialized dishes of Victorian era: ice cream sets, berry sets, lemonade sets, etc.
