So what is a finger bowl? It is essentially a little bowl of water that you dip your fingers in to clean them at the dinner table.
Some reference books claim they have been used on and off from medieval times until now, but I can barely find any references to them in etiquette books before 1900. So I believe finger bowls as we imagine them must have been a late Victorian/Gilded Age invention.
Finger bowls are always served with the dessert course. In fact, as the change of plates and silverware for the dessert course is brought out, the finger bowl is actual on top of a doily on top of the dessert plate. The diner then removes the finger bowl from the plate and places it (and the doily!) to the front and slightly left of the place setting. However, if there is no silverware on the plate with the finger bowl, it signals that there is no dessert and then the finger bowl is left on the plate.
To use the finger bowl, you gently dip the very tips of your fingers into the water and then dry them off with your napkin. You may also dab a bit of the water onto your lips with your fingers and then pat dry with the napkin. You are not really supposed to be washing your hands, merely giving a polite impression of cleanliness.
A more serious, soapy bowl of water may be given after eating messy foods such as lobster. Of course, nowadays we have those handy packets of wet wipes that rib joints pass out- not quite as elegant but definitely more practical.
Of course my favorite finger bowl story is the one where the impolite rube is at a fancy society dinner and drinks the contents of the bowl. His hostess (often said to be Queen Victoria) promptly drinks her finger bowl water also, so that he doesn’t feel that he has done something wrong. A perfect example of the spirit in which etiquette exists: to make everyone feel comfortable.
Interestingly, despite the fact that no one actually uses finger bowls anymore, every contemporary book of dining etiquette mentions how to use them. I’ve eaten at a lot of fancy places and have never seen them. If you have, please tell me where in the comments!