This question was posed to me by a lady as I sat sipping a drink in the little restaurant area of the New York Ceramics Fair. Stirrup cups were big at the fair this year. A handsome example adorned the cover of the fair catalog, and several dealers had lovely examples on display. This lady, who apparently pops into the fair each year and presumably buys on occasion, had looked at all the stirrup cups on display....and had been too intimidated to ask what purpose they served.
It was the "parting cup" given to departing guests, ready to leave with their feet in the stirrups. It was also the vehicle for serving a drink (usually sherry or port) served prior to a traditional foxhunt.
So there it is in a nutshell. But the lady bothered me. Why had she hesitated to ask one of the very knowledgable dealers ? The bunch at the fair are particularly friendly and enthusiastic about their stock. Perhaps she had been rebuffed in the past. So get over it. I have had my share of put-downs....and now I just love it when someone tries. I have all the pat answers ready. Never hesitate to ask a question. It's the only way you will learn.
Another factoid that not everyone knows is the purpose of a spill holder, sometimes called a spill vase
While I am no Martha Stewart, I use my spill holders for more decorative purposes. I put fresh flowers in them. The flowers are placed in those little florists tubes that hold water. They can last for quite a few days...and they look so pretty when you entertain. Of course, if you want a place to store your chop sticks, you now know you should go out and buy a spill holder!