At first, I thought you folks were talking about something similar to a parfait, but after looking it up on Wikipedia, I can see you're not.
Def.: A verrine is a confection, originally from France, made by layering ingredients in a small glass. It can be either sweet or savoury, making a dessert or snack.
It sounds very interesting and a great party idea!
"Food is our common ground, a universal experience." - James Beard